Common Graphs¶

All graphs in Sage can be built through the graphs object. In order to build a complete graph on 15 elements, one can do:

sage: g = graphs.CompleteGraph(15)


To get a path with 4 vertices, and the house graph:

sage: p = graphs.PathGraph(4)
sage: h = graphs.HouseGraph()


More interestingly, one can get the list of all graphs that Sage knows how to build by typing graphs. in Sage and then hitting tab.

Basic structures

Small Graphs

A small graph is just a single graph and has no parameter influencing the number of edges or vertices.

Platonic solids (ordered ascending by number of vertices)

Families of graphs

A family of graph is an infinite set of graphs which can be indexed by fixed number of parameters, e.g. two integer parameters. (A method whose name starts with a small letter does not return a single graph object but a graph iterator or a list of graphs or …)

Graphs from classical geometries over finite fields

A number of classes of graphs related to geometries over finite fields and quadrics and Hermitean varieties there.

Chessboard Graphs

Intersection graphs

These graphs are generated by geometric representations. The objects of the representation correspond to the graph vertices and the intersections of objects yield the graph edges.

Random graphs

Graphs with a given degree sequence

Miscellaneous

AUTHORS:

• Robert Miller (2006-11-05): initial version, empty, random, petersen
• Emily Kirkman (2006-11-12): basic structures, node positioning for all constructors
• Emily Kirkman (2006-11-19): docstrings, examples
• William Stein (2006-12-05): Editing.
• Robert Miller (2007-01-16): Cube generation and plotting
• Emily Kirkman (2007-01-16): more basic structures, docstrings
• Emily Kirkman (2007-02-14): added more named graphs
• Robert Miller (2007-06-08-11): Platonic solids, random graphs, graphs with a given degree sequence, random directed graphs
• Robert Miller (2007-10-24): Isomorph free exhaustive generation
• Nathann Cohen (2009-08-12): WorldMap
• Michael Yurko (2009-9-01): added hyperstar, (n,k)-star, n-star, and bubblesort graphs
• Anders Jonsson (2009-10-15): added generalized Petersen graphs
• Harald Schilly and Yann Laigle-Chapuy (2010-03-24): added Fibonacci Tree
• Jason Grout (2010-06-04): cospectral_graphs
• Edward Scheinerman (2010-08-11): RandomTree
• Ed Scheinerman (2010-08-21): added Grotzsch graph and Mycielski graphs
• Ed Scheinerman (2010-11-15): added RandomTriangulation
• Minh Van Nguyen (2010-11-26): added more named graphs
• Keshav Kini (2011-02-16): added Shrikhande and Dyck graphs
• David Coudert (2012-02-10): new RandomGNP generator
• David Coudert (2012-08-02): added chessboard graphs: Queen, King, Knight, Bishop, and Rook graphs
• Nico Van Cleemput (2013-05-26): added fullerenes
• Nico Van Cleemput (2013-07-01): added benzenoids
• Birk Eisermann (2013-07-29): new section ‘intersection graphs’, added (random, bounded) tolerance graphs
• Marco Cognetta (2016-03-03): added TuranGraph

Functions and methods¶

class sage.graphs.graph_generators.GraphGenerators

A class consisting of constructors for several common graphs, as well as orderly generation of isomorphism class representatives. See the module's help for a list of supported constructors.

A list of all graphs and graph structures (other than isomorphism class representatives) in this database is available via tab completion. Type “graphs.” and then hit the tab key to see which graphs are available.

The docstrings include educational information about each named graph with the hopes that this class can be used as a reference.

For all the constructors in this class (except the octahedral, dodecahedral, random and empty graphs), the position dictionary is filled to override the spring-layout algorithm.

ORDERLY GENERATION:

graphs(vertices, property=lambda x: True, augment='edges', size=None)


This syntax accesses the generator of isomorphism class representatives. Iterates over distinct, exhaustive representatives.

Also: see the use of the nauty package for generating graphs at the nauty_geng() method.

INPUT:

• vertices – a natural number or None to infinitely generate bigger and bigger graphs.
• property – (default: lambda x: True) any property to be tested on graphs before generation, but note that in general the graphs produced are not the same as those produced by using the property function to filter a list of graphs produced by using the lambda x: True default. The generation process assumes the property has certain characteristics set by the augment argument, and only in the case of inherited properties such that all subgraphs of the relevant kind (for augment='edges' or augment='vertices') of a graph with the property also possess the property will there be no missing graphs. (The property argument is ignored if degree_sequence is specified.)
• augment – (default: 'edges') possible values:
• 'edges' – augments a fixed number of vertices by adding one edge. In this case, all graphs on exactly n=vertices are generated. If for any graph G satisfying the property, every subgraph, obtained from G by deleting one edge but not the vertices incident to that edge, satisfies the property, then this will generate all graphs with that property. If this does not hold, then all the graphs generated will satisfy the property, but there will be some missing.
• 'vertices' – augments by adding a vertex and edges incident to that vertex. In this case, all graphs up to n=vertices are generated. If for any graph G satisfying the property, every subgraph, obtained from G by deleting one vertex and only edges incident to that vertex, satisfies the property, then this will generate all graphs with that property. If this does not hold, then all the graphs generated will satisfy the property, but there will be some missing.
• size – (default: None) the size of the graph to be generated.
• degree_sequence – (default: None) a sequence of non-negative integers, or None. If specified, the generated graphs will have these integers for degrees. In this case, property and size are both ignored.
• loops – (default: False) whether to allow loops in the graph or not.
• implementation – (default: 'c_graph') which underlying implementation to use (see Graph?).
• sparse – (default: True) ignored if implementation is not 'c_graph'.
• copy (boolean) – If set to True (default) this method makes copies of the graphs before returning them. If set to False the method returns the graph it is working on. The second alternative is faster, but modifying any of the graph instances returned by the method may break the function’s behaviour, as it is using these graphs to compute the next ones: only use copy = False when you stick to reading the graphs returned.

EXAMPLES:

Print graphs on 3 or less vertices:

sage: for G in graphs(3, augment='vertices'):
....:     print(G)
Graph on 0 vertices
Graph on 1 vertex
Graph on 2 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 2 vertices
Graph on 3 vertices


Note that we can also get graphs with underlying Cython implementation:

sage: for G in graphs(3, augment='vertices', implementation='c_graph'):
....:     print(G)
Graph on 0 vertices
Graph on 1 vertex
Graph on 2 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 2 vertices
Graph on 3 vertices


Print graphs on 3 vertices.

sage: for G in graphs(3):
....:    print(G)
Graph on 3 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 3 vertices


Generate all graphs with 5 vertices and 4 edges.

sage: L = graphs(5, size=4)
sage: len(list(L))
6


Generate all graphs with 5 vertices and up to 4 edges.

sage: L = list(graphs(5, lambda G: G.size() <= 4))
sage: len(L)
14
sage: graphs_list.show_graphs(L) # long time


Generate all graphs with up to 5 vertices and up to 4 edges.

sage: L = list(graphs(5, lambda G: G.size() <= 4, augment='vertices'))
sage: len(L)
31
sage: graphs_list.show_graphs(L)              # long time


Generate all graphs with degree at most 2, up to 6 vertices.

sage: property = lambda G: ( max([G.degree(v) for v in G] + ) <= 2 )
sage: L = list(graphs(6, property, augment='vertices'))
sage: len(L)
45


Generate all bipartite graphs on up to 7 vertices: (see OEIS sequence A033995)

sage: L = list( graphs(7, lambda G: G.is_bipartite(), augment='vertices') )
sage: [len([g for g in L if g.order() == i]) for i in [1..7]]
[1, 2, 3, 7, 13, 35, 88]


Generate all bipartite graphs on exactly 7 vertices:

sage: L = list( graphs(7, lambda G: G.is_bipartite()) )
sage: len(L)
88


Generate all bipartite graphs on exactly 8 vertices:

sage: L = list( graphs(8, lambda G: G.is_bipartite()) ) # long time
sage: len(L)                                            # long time
303


Remember that the property argument does not behave as a filter, except for appropriately inheritable properties:

sage: property = lambda G: G.is_vertex_transitive()
sage: len(list(graphs(4, property)))
1
sage: sum(1 for g in graphs(4) if property(g))
4

sage: property = lambda G: G.is_bipartite()
sage: len(list(graphs(4, property)))
7
sage: sum(1 for g in graphs(4) if property(g))
7


Generate graphs on the fly: (see OEIS sequence A000088)

sage: for i in range(7):
....:     print(len(list(graphs(i))))
1
1
2
4
11
34
156


Generate all simple graphs, allowing loops: (see OEIS sequence A000666)

sage: L = list(graphs(5,augment='vertices',loops=True))               # long time
sage: for i in [0..5]:  # long time
....:     print((i, len([g for g in L if g.order() == i]))) # long time
(0, 1)
(1, 2)
(2, 6)
(3, 20)
(4, 90)
(5, 544)


Generate all graphs with a specified degree sequence (see OEIS sequence A002851):

sage: for i in [4,6,8]:  # long time (4s on sage.math, 2012)
....:     print((i, len([g for g in graphs(i, degree_sequence=*i) if g.is_connected()])))
(4, 1)
(6, 2)
(8, 5)
sage: for i in [4,6,8]:  # long time (7s on sage.math, 2012)
....:     print((i, len([g for g in graphs(i, augment='vertices', degree_sequence=*i) if g.is_connected()])))
(4, 1)
(6, 2)
(8, 5)

sage: print((10, len([g for g in graphs(10,degree_sequence=*10) if g.is_connected()]))) # not tested
(10, 19)


Make sure that the graphs are really independent and the generator survives repeated vertex removal (trac ticket #8458):

sage: for G in graphs(3):
....:     G.delete_vertex(0)
....:     print(G.order())
2
2
2
2


REFERENCE:

• Brendan D. McKay, Isomorph-Free Exhaustive generation. Journal of Algorithms, Volume 26, Issue 2, February 1998, pages 306-324.
static AffineOrthogonalPolarGraph(d, q, sign='+')

Returns the affine polar graph $$VO^+(d,q),VO^-(d,q)$$ or $$VO(d,q)$$.

Affine Polar graphs are built from a $$d$$-dimensional vector space over $$F_q$$, and a quadratic form which is hyperbolic, elliptic or parabolic according to the value of sign.

Note that $$VO^+(d,q),VO^-(d,q)$$ are strongly regular graphs, while $$VO(d,q)$$ is not.

For more information on Affine Polar graphs, see Affine Polar Graphs page of Andries Brouwer’s website.

INPUT:

• d (integer) – d must be even if sign is not None, and odd otherwise.
• q (integer) – a power of a prime number, as $$F_q$$ must exist.
• sign – must be equal to "+", "-", or None to compute (respectively) $$VO^+(d,q),VO^-(d,q)$$ or $$VO(d,q)$$. By default sign="+".

Note

The graph $$VO^\epsilon(d,q)$$ is the graph induced by the non-neighbors of a vertex in an Orthogonal Polar Graph $$O^\epsilon(d+2,q)$$.

EXAMPLES:

The Brouwer-Haemers graph is isomorphic to $$VO^-(4,3)$$:

sage: g = graphs.AffineOrthogonalPolarGraph(4,3,"-")
sage: g.is_isomorphic(graphs.BrouwerHaemersGraph())
True


Some examples from Brouwer’s table or strongly regular graphs:

sage: g = graphs.AffineOrthogonalPolarGraph(6,2,"-"); g
Affine Polar Graph VO^-(6,2): Graph on 64 vertices
sage: g.is_strongly_regular(parameters=True)
(64, 27, 10, 12)
sage: g = graphs.AffineOrthogonalPolarGraph(6,2,"+"); g
Affine Polar Graph VO^+(6,2): Graph on 64 vertices
sage: g.is_strongly_regular(parameters=True)
(64, 35, 18, 20)


When sign is None:

sage: g = graphs.AffineOrthogonalPolarGraph(5,2,None); g
Affine Polar Graph VO^-(5,2): Graph on 32 vertices
sage: g.is_strongly_regular(parameters=True)
False
sage: g.is_regular()
True
sage: g.is_vertex_transitive()
True

static AfricaMap(continental=False, year=2018)

Return African states as a graph of common border.

“African state” here is defined as an independent state having the capital city in Africa. The graph has an edge between those countries that have common land border.

INPUT:

• continental, a Boolean – if set, only return states in the continental Africa
• year – reserved for future use

EXAMPLES:

sage: Africa = graphs.AfricaMap(); Africa
Africa Map: Graph on 54 vertices
sage: sorted(Africa.neighbors('Libya'))
['Algeria', 'Chad', 'Egypt', 'Niger', 'Sudan', 'Tunisia']

sage: cont_Africa = graphs.AfricaMap(continental=True)
sage: cont_Africa.order()
48
False

static AhrensSzekeresGeneralizedQuadrangleGraph(q, dual=False)

Return the collinearity graph of the generalized quadrangle $$AS(q)$$, or of its dual

Let $$q$$ be an odd prime power. $$AS(q)$$ is a generalized quadrangle [GQwiki] of order $$(q-1,q+1)$$, see 3.1.5 in [PT09]. Its points are elements of $$F_q^3$$, and lines are sets of size $$q$$ of the form

• $$\{ (\sigma, a, b) \mid \sigma\in F_q \}$$
• $$\{ (a, \sigma, b) \mid \sigma\in F_q \}$$
• $$\{ (c \sigma^2 - b \sigma + a, -2 c \sigma + b, \sigma) \mid \sigma\in F_q \}$$,

where $$a$$, $$b$$, $$c$$ are arbitrary elements of $$F_q$$.

INPUT:

• q – a power of an odd prime number
• dual – if False (default), return the collinearity graph of $$AS(q)$$. Otherwise return the collinearity graph of the dual $$AS(q)$$.

EXAMPLES:

sage: g=graphs.AhrensSzekeresGeneralizedQuadrangleGraph(5); g
AS(5); GQ(4, 6): Graph on 125 vertices
sage: g.is_strongly_regular(parameters=True)
(125, 28, 3, 7)
AS(5)*; GQ(6, 4): Graph on 175 vertices
sage: g.is_strongly_regular(parameters=True)
(175, 30, 5, 5)


REFERENCE:

 [GQwiki] (1, 2, 3) Wikipedia article Generalized_quadrangle
 [PT09] (1, 2, 3) S. Payne, J. A. Thas. Finite generalized quadrangles. European Mathematical Society, 2nd edition, 2009.
static AztecDiamondGraph(n)

Return the Aztec Diamond graph of order n.

EXAMPLES:

sage: graphs.AztecDiamondGraph(2)
Aztec Diamond graph of order 2

sage: [graphs.AztecDiamondGraph(i).num_verts() for i in range(8)]
[0, 4, 12, 24, 40, 60, 84, 112]

sage: [graphs.AztecDiamondGraph(i).num_edges() for i in range(8)]
[0, 4, 16, 36, 64, 100, 144, 196]

sage: G = graphs.AztecDiamondGraph(3)
sage: sum(1 for p in G.perfect_matchings())
64

static Balaban10Cage(embedding=1)

Return the Balaban 10-cage.

The Balaban 10-cage is a 3-regular graph with 70 vertices and 105 edges. See the Wikipedia article Balaban_10-cage.

The default embedding gives a deeper understanding of the graph’s automorphism group. It is divided into 4 layers (each layer being a set of points at equal distance from the drawing’s center). From outside to inside:

• L1: The outer layer (vertices which are the furthest from the origin) is actually the disjoint union of two cycles of length 10.
• L2: The second layer is an independent set of 20 vertices.
• L3: The third layer is a matching on 10 vertices.
• L4: The inner layer (vertices which are the closest from the origin) is also the disjoint union of two cycles of length 10.

This graph is not vertex-transitive, and its vertices are partitioned into 3 orbits: L2, L3, and the union of L1 of L4 whose elements are equivalent.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to be either 1 or 2.

EXAMPLES:

sage: g = graphs.Balaban10Cage()
sage: g.girth()
10
sage: g.chromatic_number()
2
sage: g.diameter()
6
sage: g.is_hamiltonian()
True
sage: g.show(figsize=[10,10])   # long time

static Balaban11Cage(embedding=1)

Return the Balaban 11-cage.

INPUT:

• embedding – three embeddings are available, and can be selected by setting embedding to be 1, 2, or 3.
• The first embedding is the one appearing on page 9 of the Fifth Annual Graph Drawing Contest report [EMMN1998]. It separates vertices based on their eccentricity (see eccentricity()).
• The second embedding has been produced just for Sage and is meant to emphasize the automorphism group’s 6 orbits.
• The last embedding is the default one produced by the LCFGraph() constructor.

Note

The vertex labeling changes according to the value of embedding=1.

EXAMPLES:

Basic properties:

sage: g = graphs.Balaban11Cage()
sage: g.order()
112
sage: g.size()
168
sage: g.girth()
11
sage: g.diameter()
8
sage: g.automorphism_group().cardinality()
64


Our many embeddings:

sage: g1 = graphs.Balaban11Cage(embedding=1)
sage: g2 = graphs.Balaban11Cage(embedding=2)
sage: g3 = graphs.Balaban11Cage(embedding=3)
sage: g1.show(figsize=[10,10])   # long time
sage: g2.show(figsize=[10,10])   # long time
sage: g3.show(figsize=[10,10])   # long time


Proof that the embeddings are the same graph:

sage: g1.is_isomorphic(g2) # g2 and g3 are obviously isomorphic
True

static BalancedTree(r, h)

Returns the perfectly balanced tree of height $$h \geq 1$$, whose root has degree $$r \geq 2$$.

The number of vertices of this graph is $$1 + r + r^2 + \cdots + r^h$$, that is, $$\frac{r^{h+1} - 1}{r - 1}$$. The number of edges is one less than the number of vertices.

INPUT:

• r – positive integer $$\geq 2$$. The degree of the root node.
• h – positive integer $$\geq 1$$. The height of the balanced tree.

OUTPUT:

The perfectly balanced tree of height $$h \geq 1$$ and whose root has degree $$r \geq 2$$. A NetworkXError is returned if $$r < 2$$ or $$h < 1$$.

ALGORITHM:

Uses NetworkX.

EXAMPLES:

A balanced tree whose root node has degree $$r = 2$$, and of height $$h = 1$$, has order 3 and size 2:

sage: G = graphs.BalancedTree(2, 1); G
Balanced tree: Graph on 3 vertices
sage: G.order(); G.size()
3
2
sage: r = 2; h = 1
sage: v = 1 + r
sage: v; v - 1
3
2


Plot a balanced tree of height 5, whose root node has degree $$r = 3$$:

sage: G = graphs.BalancedTree(3, 5)
sage: G.show()   # long time


A tree is bipartite. If its vertex set is finite, then it is planar.

sage: r = randint(2, 5); h = randint(1, 7)
sage: T = graphs.BalancedTree(r, h)
sage: T.is_bipartite()
True
sage: T.is_planar()
True
sage: v = (r^(h + 1) - 1) / (r - 1)
sage: T.order() == v
True
sage: T.size() == v - 1
True

static BarbellGraph(n1, n2)

Returns a barbell graph with 2*n1 + n2 nodes. The argument n1 must be greater than or equal to 2.

A barbell graph is a basic structure that consists of a path graph of order n2 connecting two complete graphs of order n1 each.

INPUT:

• n1 – integer $$\geq 2$$. The order of each of the two complete graphs.
• n2 – nonnegative integer. The order of the path graph connecting the two complete graphs.

OUTPUT:

A barbell graph of order 2*n1 + n2. A ValueError is returned if n1 < 2 or n2 < 0.

PLOTTING:

Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each barbell graph will be displayed with the two complete graphs in the lower-left and upper-right corners, with the path graph connecting diagonally between the two. Thus the n1-th node will be drawn at a 45 degree angle from the horizontal right center of the first complete graph, and the n1 + n2 + 1-th node will be drawn 45 degrees below the left horizontal center of the second complete graph.

EXAMPLES:

Construct and show a barbell graph Bar = 4, Bells = 9:

sage: g = graphs.BarbellGraph(9, 4); g
Barbell graph: Graph on 22 vertices
sage: g.show() # long time


An n1 >= 2, n2 >= 0 barbell graph has order 2*n1 + n2. It has the complete graph on n1 vertices as a subgraph. It also has the path graph on n2 vertices as a subgraph.

sage: n1 = randint(2, 2*10^2)
sage: n2 = randint(0, 2*10^2)
sage: g = graphs.BarbellGraph(n1, n2)
sage: v = 2*n1 + n2
sage: g.order() == v
True
sage: K_n1 = graphs.CompleteGraph(n1)
sage: P_n2 = graphs.PathGraph(n2)
sage: s_K = g.subgraph_search(K_n1, induced=True)
sage: s_P = g.subgraph_search(P_n2, induced=True)
sage: K_n1.is_isomorphic(s_K)
True
sage: P_n2.is_isomorphic(s_P)
True

static BidiakisCube()

Return the Bidiakis cube.

EXAMPLES:

The Bidiakis cube is a 3-regular graph having 12 vertices and 18 edges. This means that each vertex has a degree of 3.

sage: g = graphs.BidiakisCube(); g
Bidiakis cube: Graph on 12 vertices
sage: g.show()  # long time
sage: g.order()
12
sage: g.size()
18
sage: g.is_regular(3)
True


It is a Hamiltonian graph with diameter 3 and girth 4:

sage: g.is_hamiltonian()
True
sage: g.diameter()
3
sage: g.girth()
4


It is a planar graph with characteristic polynomial $$(x - 3) (x - 2) (x^4) (x + 1) (x + 2) (x^2 + x - 4)^2$$ and chromatic number 3:

sage: g.is_planar()
True
sage: bool(g.characteristic_polynomial() == expand((x - 3) * (x - 2) * (x^4) * (x + 1) * (x + 2) * (x^2 + x - 4)^2))
True
sage: g.chromatic_number()
3

static BiggsSmithGraph(embedding=1)

Return the Biggs-Smith graph.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to be 1 or 2.

EXAMPLES:

Basic properties:

sage: g = graphs.BiggsSmithGraph()
sage: g.order()
102
sage: g.size()
153
sage: g.girth()
9
sage: g.diameter()
7
sage: g.automorphism_group().cardinality() # long time
2448
sage: g.show(figsize=[10, 10])   # long time


The other embedding:

sage: graphs.BiggsSmithGraph(embedding=2).show() # long time

static BishopGraph(dim_list, radius=None, relabel=False)

Returns the $$d$$-dimensional Bishop Graph with prescribed dimensions.

The 2-dimensional Bishop Graph of parameters $$n$$ and $$m$$ is a graph with $$nm$$ vertices in which each vertex represents a square in an $$n \times m$$ chessboard, and each edge corresponds to a legal move by a bishop.

The $$d$$-dimensional Bishop Graph with $$d >= 2$$ has for vertex set the cells of a $$d$$-dimensional grid with prescribed dimensions, and each edge corresponds to a legal move by a bishop in any pairs of dimensions.

The Bishop Graph is not connected.

INPUT:

• dim_list – an iterable object (list, set, dict) providing the dimensions $$n_1, n_2, \ldots, n_d$$, with $$n_i \geq 1$$, of the chessboard.
• radius – (default: None) by setting the radius to a positive integer, one may decrease the power of the bishop to at most radius steps.
• relabel – (default: False) a boolean set to True if vertices must be relabeled as integers.

EXAMPLES:

The (n,m)-Bishop Graph is not connected:

sage: G = graphs.BishopGraph( [3, 4] )
sage: G.is_connected()
False


The Bishop Graph can be obtained from Knight Graphs:

sage: for d in range(3,12):   # long time
....:     H = Graph()
....:     for r in range(1,d+1):
....:         if not B.is_isomorphic(H):
....:            print("that's not good!")

static BlanusaFirstSnarkGraph()

Return the first Blanusa Snark Graph.

The Blanusa graphs are two snarks on 18 vertices and 27 edges. For more information on them, see the Wikipedia article Blanusa_snarks.

EXAMPLES:

sage: g = graphs.BlanusaFirstSnarkGraph()
sage: g.order()
18
sage: g.size()
27
sage: g.diameter()
4
sage: g.girth()
5
sage: g.automorphism_group().cardinality()
8

static BlanusaSecondSnarkGraph()

Return the second Blanusa Snark Graph.

The Blanusa graphs are two snarks on 18 vertices and 27 edges. For more information on them, see the Wikipedia article Blanusa_snarks.

EXAMPLES:

sage: g = graphs.BlanusaSecondSnarkGraph()
sage: g.order()
18
sage: g.size()
27
sage: g.diameter()
4
sage: g.girth()
5
sage: g.automorphism_group().cardinality()
4

static BrinkmannGraph()

Return the Brinkmann graph.

EXAMPLES:

The Brinkmann graph is a 4-regular graph having 21 vertices and 42 edges. This means that each vertex has degree 4.

sage: G = graphs.BrinkmannGraph(); G
Brinkmann graph: Graph on 21 vertices
sage: G.show()  # long time
sage: G.order()
21
sage: G.size()
42
sage: G.is_regular(4)
True


It is an Eulerian graph with radius 3, diameter 3, and girth 5.

sage: G.is_eulerian()
True
3
sage: G.diameter()
3
sage: G.girth()
5


The Brinkmann graph is also Hamiltonian with chromatic number 4:

sage: G.is_hamiltonian()
True
sage: G.chromatic_number()
4


Its automorphism group is isomorphic to $$D_7$$:

sage: ag = G.automorphism_group()
sage: ag.is_isomorphic(DihedralGroup(7))
True

static BrouwerHaemersGraph()

Return the Brouwer-Haemers Graph.

The Brouwer-Haemers is the only strongly regular graph of parameters $$(81,20,1,6)$$. It is build in Sage as the Affine Orthogonal graph $$VO^-(6,3)$$. For more information on this graph, see its corresponding page on Andries Brouwer’s website.

EXAMPLES:

sage: g = graphs.BrouwerHaemersGraph()
sage: g
Brouwer-Haemers: Graph on 81 vertices


It is indeed strongly regular with parameters $$(81,20,1,6)$$:

sage: g.is_strongly_regular(parameters = True) # long time
(81, 20, 1, 6)


Its has as eigenvalues $$20,2$$ and $$-7$$:

sage: set(g.spectrum()) == {20,2,-7}
True

static BubbleSortGraph(n)

Returns the bubble sort graph $$B(n)$$.

The vertices of the bubble sort graph are the set of permutations on $$n$$ symbols. Two vertices are adjacent if one can be obtained from the other by swapping the labels in the $$i$$-th and $$(i+1)$$-th positions for $$1 \leq i \leq n-1$$. In total, $$B(n)$$ has order $$n!$$. Swapping two labels as described previously corresponds to multiplying on the right the permutation corresponding to the node by an elementary transposition in the SymmetricGroup.

The bubble sort graph is the underlying graph of the permutahedron().

INPUT:

• n – positive integer. The number of symbols to permute.

OUTPUT:

The bubble sort graph $$B(n)$$ on $$n$$ symbols. If $$n < 1$$, a ValueError is returned.

EXAMPLES:

sage: g = graphs.BubbleSortGraph(4); g
Bubble sort: Graph on 24 vertices
sage: g.plot() # long time
Graphics object consisting of 61 graphics primitives


The bubble sort graph on $$n = 1$$ symbol is the trivial graph $$K_1$$:

sage: graphs.BubbleSortGraph(1)
Bubble sort: Graph on 1 vertex


If $$n \geq 1$$, then the order of $$B(n)$$ is $$n!$$:

sage: n = randint(1, 8)
sage: g = graphs.BubbleSortGraph(n)
sage: g.order() == factorial(n)
True


AUTHORS:

• Michael Yurko (2009-09-01)
static BuckyBall()

Create the Bucky Ball graph.

This graph is a 3-regular 60-vertex planar graph. Its vertices and edges correspond precisely to the carbon atoms and bonds in buckminsterfullerene. When embedded on a sphere, its 12 pentagon and 20 hexagon faces are arranged exactly as the sections of a soccer ball.

EXAMPLES:

The Bucky Ball is planar.

sage: g = graphs.BuckyBall()
sage: g.is_planar()
True


The Bucky Ball can also be created by extracting the 1-skeleton of the Bucky Ball polyhedron, but this is much slower.

sage: g = polytopes.buckyball().vertex_graph()
sage: g.remove_loops()
sage: h = graphs.BuckyBall()
sage: g.is_isomorphic(h)
True


The graph is returned along with an attractive embedding.

sage: g = graphs.BuckyBall()
sage: g.plot(vertex_labels=False, vertex_size=10).show() # long time

static BullGraph()

Returns a bull graph with 5 nodes.

A bull graph is named for its shape. It’s a triangle with horns. See the Wikipedia article Bull_graph for more information.

PLOTTING:

Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the bull graph is drawn as a triangle with the first node (0) on the bottom. The second and third nodes (1 and 2) complete the triangle. Node 3 is the horn connected to 1 and node 4 is the horn connected to node 2.

EXAMPLES:

Construct and show a bull graph:

sage: g = graphs.BullGraph(); g
Bull graph: Graph on 5 vertices
sage: g.show() # long time


The bull graph has 5 vertices and 5 edges. Its radius is 2, its diameter 3, and its girth 3. The bull graph is planar with chromatic number 3 and chromatic index also 3.

sage: g.order(); g.size()
5
5
2
3
3
sage: g.chromatic_number()
3


The bull graph has chromatic polynomial $$x(x - 2)(x - 1)^3$$ and Tutte polynomial $$x^4 + x^3 + x^2 y$$. Its characteristic polynomial is $$x(x^2 - x - 3)(x^2 + x - 1)$$, which follows from the definition of characteristic polynomials for graphs, i.e. $$\det(xI - A)$$, where $$x$$ is a variable, $$A$$ the adjacency matrix of the graph, and $$I$$ the identity matrix of the same dimensions as $$A$$.

sage: chrompoly = g.chromatic_polynomial()
sage: bool(expand(x * (x - 2) * (x - 1)^3) == chrompoly)
True
sage: charpoly = g.characteristic_polynomial()
[0 1 1 0 0]
[1 0 1 1 0]
[1 1 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
sage: Id = identity_matrix(ZZ, M.nrows())
sage: D = x*Id - M
sage: bool(D.determinant() == charpoly)
True
sage: bool(expand(x * (x^2 - x - 3) * (x^2 + x - 1)) == charpoly)
True

static ButterflyGraph()

Returns the butterfly graph.

Let $$C_3$$ be the cycle graph on 3 vertices. The butterfly or bowtie graph is obtained by joining two copies of $$C_3$$ at a common vertex, resulting in a graph that is isomorphic to the friendship graph $$F_2$$. See the Wikipedia article Butterfly_graph for more information.

EXAMPLES:

The butterfly graph is a planar graph on 5 vertices and having 6 edges.

sage: G = graphs.ButterflyGraph(); G
Butterfly graph: Graph on 5 vertices
sage: G.show()  # long time
sage: G.is_planar()
True
sage: G.order()
5
sage: G.size()
6


It has diameter 2, girth 3, and radius 1.

sage: G.diameter()
2
sage: G.girth()
3
1


The butterfly graph is Eulerian, with chromatic number 3.

sage: G.is_eulerian()
True
sage: G.chromatic_number()
3

static CaiFurerImmermanGraph(G, twisted=False)

Return the a Cai-Furer-Immerman graph from $$G$$, possibly a twisted one, and a partition of its nodes.

A Cai-Furer-Immerman graph from/on $$G$$ is a graph created by applying the transformation described in [CFI1992] on a graph $$G$$, that is substituting every vertex v in $$G$$ with a Furer gadget $$F(v)$$ of order d equal to the degree of the vertex, and then substituting every edge $$(v,u)$$ in $$G$$ with a pair of edges, one connecting the two “a” nodes of $$F(v)$$ and $$F(u)$$ and the other their two “b” nodes. The returned coloring of the vertices is made by the union of the colorings of each single Furer gadget, individualised for each vertex of $$G$$. To understand better what these “a” and “b” nodes are, see the documentation on Furer gadgets.

Furthermore, this method can apply what is described in the paper mentioned above as a “twist” on an edge, that is taking only one of the pairs of edges introduced in the new graph and swap two of their extremes, making each edge go from an “a” node to a “b” node. This is only doable if the original graph G is connected.

A CaiFurerImmerman graph on a graph with no balanced vertex separators smaller than s and its twisted version cannot be distinguished by k-WL for any k < s.

INPUT:

• G – An undirected graph on which to construct the
Cai-Furer-Immerman graph
• twisted – A boolean indicating if the version to construct
is a twisted one or not

OUTPUT:

• H – The Cai-Furer-Immerman graph on G
• coloring – A list of list of vertices, representing the
partition induced by the coloring on H

EXAMPLES:

CaiFurerImmerman graph with no balanced vertex separator smaller than 2

sage: G = graphs.CycleGraph(4)
sage: CFI, p = graphs.CaiFurerImmermanGraph(G)
sage: sorted(CFI, key=str)
[(0, ()), (0, (0, 'a')), (0, (0, 'b')), (0, (0, 1)), (0, (1, 'a')),
(0, (1, 'b')), (1, ()), (1, (0, 'a')), (1, (0, 'b')), (1, (0, 1)),
(1, (1, 'a')), (1, (1, 'b')), (2, ()), (2, (0, 'a')), (2, (0, 'b')),
(2, (0, 1)), (2, (1, 'a')), (2, (1, 'b')), (3, ()), (3, (0, 'a')),
(3, (0, 'b')), (3, (0, 1)), (3, (1, 'a')), (3, (1, 'b'))]
sage: sorted(CFI.edge_iterator(), key=str)
[((0, ()), (0, (0, 'b')), None),
((0, ()), (0, (1, 'b')), None),
((0, (0, 'a')), (1, (0, 'a')), None),
((0, (0, 'b')), (1, (0, 'b')), None),
((0, (0, 1)), (0, (0, 'a')), None),
((0, (0, 1)), (0, (1, 'a')), None),
((0, (1, 'a')), (3, (0, 'a')), None),
((0, (1, 'b')), (3, (0, 'b')), None),
((1, ()), (1, (0, 'b')), None),
((1, ()), (1, (1, 'b')), None),
((1, (0, 1)), (1, (0, 'a')), None),
((1, (0, 1)), (1, (1, 'a')), None),
((1, (1, 'a')), (2, (0, 'a')), None),
((1, (1, 'b')), (2, (0, 'b')), None),
((2, ()), (2, (0, 'b')), None),
((2, ()), (2, (1, 'b')), None),
((2, (0, 1)), (2, (0, 'a')), None),
((2, (0, 1)), (2, (1, 'a')), None),
((2, (1, 'a')), (3, (1, 'a')), None),
((2, (1, 'b')), (3, (1, 'b')), None),
((3, ()), (3, (0, 'b')), None),
((3, ()), (3, (1, 'b')), None),
((3, (0, 1)), (3, (0, 'a')), None),
((3, (0, 1)), (3, (1, 'a')), None)]

static CameronGraph()

Return the Cameron graph.

The Cameron graph is strongly regular with parameters $$v = 231, k = 30, \lambda = 9, \mu = 3$$.

EXAMPLES:

sage: g = graphs.CameronGraph()
sage: g.order()
231
sage: g.size()
3465
sage: g.is_strongly_regular(parameters = True) # long time
(231, 30, 9, 3)

static Cell120()

Return the 120-Cell graph.

This is the adjacency graph of the 120-cell. It has 600 vertices and 1200 edges. For more information, see the Wikipedia article 120-cell.

EXAMPLES:

sage: g = graphs.Cell120()      # long time
sage: g.size()                  # long time
1200
sage: g.is_regular(4)           # long time
True
sage: g.is_vertex_transitive()  # long time
True

static Cell600(embedding=1)

Return the 600-Cell graph.

This is the adjacency graph of the 600-cell. It has 120 vertices and 720 edges. For more information, see the Wikipedia article 600-cell.

INPUT:

• embedding (1 (default) or 2) – two different embeddings for a plot.

EXAMPLES:

sage: g = graphs.Cell600()      # long time
sage: g.size()                  # long time
720
sage: g.is_regular(12)          # long time
True
sage: g.is_vertex_transitive()  # long time
True

static ChessboardGraphGenerator(dim_list, rook=True, rook_radius=None, bishop=True, bishop_radius=None, knight=True, knight_x=1, knight_y=2, relabel=False)

Returns a Graph built on a $$d$$-dimensional chessboard with prescribed dimensions and interconnections.

This function allows to generate many kinds of graphs corresponding to legal movements on a $$d$$-dimensional chessboard: Queen Graph, King Graph, Knight Graphs, Bishop Graph, and many generalizations. It also allows to avoid redundant code.

INPUT:

• dim_list – an iterable object (list, set, dict) providing the dimensions $$n_1, n_2, \ldots, n_d$$, with $$n_i \geq 1$$, of the chessboard.
• rook – (default: True) boolean value indicating if the chess piece is able to move as a rook, that is at any distance along a dimension.
• rook_radius – (default: None) integer value restricting the rook-like movements to distance at most $$rook_radius$$.
• bishop – (default: True) boolean value indicating if the chess piece is able to move like a bishop, that is along diagonals.
• bishop_radius – (default: None) integer value restricting the bishop-like movements to distance at most $$bishop_radius$$.
• knight – (default: True) boolean value indicating if the chess piece is able to move like a knight.
• knight_x – (default: 1) integer indicating the number on steps the chess piece moves in one dimension when moving like a knight.
• knight_y – (default: 2) integer indicating the number on steps the chess piece moves in the second dimension when moving like a knight.
• relabel – (default: False) a boolean set to True if vertices must be relabeled as integers.

OUTPUT:

• A Graph build on a $$d$$-dimensional chessboard with prescribed dimensions, and with edges according given parameters.
• A string encoding the dimensions. This is mainly useful for providing names to graphs.

EXAMPLES:

A $$(2,2)$$-King Graph is isomorphic to the complete graph on 4 vertices:

sage: G, _ = graphs.ChessboardGraphGenerator( [2,2] )
sage: G.is_isomorphic( graphs.CompleteGraph(4) )
True


A Rook’s Graph in 2 dimensions is isomorphic to the Cartesian product of 2 complete graphs:

sage: G, _ = graphs.ChessboardGraphGenerator( [3,4], rook=True, rook_radius=None, bishop=False, knight=False )
sage: H = ( graphs.CompleteGraph(3) ).cartesian_product( graphs.CompleteGraph(4) )
sage: G.is_isomorphic(H)
True

static ChvatalGraph()

Return the Chvatal graph.

Chvatal graph is one of the few known graphs to satisfy Grunbaum’s conjecture that for every m, n, there is an m-regular, m-chromatic graph of girth at least n. For more information, see the Wikipedia article Chv%C3%A1tal_graph.

EXAMPLES:

The Chvatal graph has 12 vertices and 24 edges. It is a 4-regular, 4-chromatic graph with radius 2, diameter 2, and girth 4.

sage: G = graphs.ChvatalGraph(); G
Chvatal graph: Graph on 12 vertices
sage: G.order(); G.size()
12
24
sage: G.degree()
[4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4]
sage: G.chromatic_number()
4
2
2
4

static CirculantGraph(n, adjacency)

Returns a circulant graph with n nodes.

A circulant graph has the property that the vertex $$i$$ is connected with the vertices $$i+j$$ and $$i-j$$ for each j in adjacency.

INPUT:

• n - number of vertices in the graph
• adjacency - the list of j values

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each circulant graph will be displayed with the first (0) node at the top, with the rest following in a counterclockwise manner.

EXAMPLES: Compare plotting using the predefined layout and networkx:

sage: import networkx
sage: n = networkx.cycle_graph(23)
sage: spring23 = Graph(n)
sage: posdict23 = graphs.CirculantGraph(23,2)
sage: spring23.show() # long time
sage: posdict23.show() # long time


We next view many cycle graphs as a Sage graphics array. First we use the CirculantGraph constructor, which fills in the position dictionary:

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.CirculantGraph(i+4, i+1)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


Compare to plotting with the spring-layout algorithm:

sage: g = []
sage: j = []
sage: for i in range(9):
....:     spr = networkx.cycle_graph(i+3)
....:     k = Graph(spr)
....:     g.append(k)
sage: for i in range(3):
....:  n = []
....:  for m in range(3):
....:      n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:  j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


Passing a 1 into adjacency should give the cycle.

sage: graphs.CirculantGraph(6,1)==graphs.CycleGraph(6)
True
sage: graphs.CirculantGraph(7,[1,3]).edges(labels=false)
[(0, 1),
(0, 3),
(0, 4),
(0, 6),
(1, 2),
(1, 4),
(1, 5),
(2, 3),
(2, 5),
(2, 6),
(3, 4),
(3, 6),
(4, 5),
(5, 6)]

static CircularLadderGraph(n)

Return a circular ladder graph with $$2 * n$$ nodes.

A Circular ladder graph is a ladder graph that is connected at the ends, i.e.: a ladder bent around so that top meets bottom. Thus it can be described as two parallel cycle graphs connected at each corresponding node pair.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the circular ladder graph is displayed as an inner and outer cycle pair, with the first $$n$$ nodes drawn on the inner circle. The first (0) node is drawn at the top of the inner-circle, moving clockwise after that. The outer circle is drawn with the $$(n+1)th node at the top, then counterclockwise as well. When n == 2$$, we rotate the outer circle by an angle of $$\pi/8$$ to ensure that all edges are visible (otherwise the 4 vertices of the graph would be placed on a single line).

EXAMPLES:

Construct and show a circular ladder graph with 26 nodes:

sage: g = graphs.CircularLadderGraph(13)
sage: g.show() # long time


Create several circular ladder graphs in a Sage graphics array:

sage: g = []
sage: j = []
sage: for i in range(9):
....:    g.append(k)
sage: for i in range(3):
....:    n = []
....:    for m in range(3):
....:        n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:    j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static ClawGraph()

Returns a claw graph.

A claw graph is named for its shape. It is actually a complete bipartite graph with (n1, n2) = (1, 3).

PLOTTING: See CompleteBipartiteGraph.

EXAMPLES: Show a Claw graph

sage: (graphs.ClawGraph()).show() # long time


Inspect a Claw graph

sage: G = graphs.ClawGraph()
sage: G
Claw graph: Graph on 4 vertices

static ClebschGraph()

Return the Clebsch graph.

EXAMPLES:

sage: g = graphs.ClebschGraph()
sage: g.automorphism_group().cardinality()
1920
sage: g.girth()
4
sage: g.chromatic_number()
4
sage: g.diameter()
2
sage: g.show(figsize=[10, 10]) # long time

static CompleteBipartiteGraph(n1, n2, set_position=True)

Return a Complete Bipartite Graph on $$n1 + n2$$ vertices.

A Complete Bipartite Graph is a graph with its vertices partitioned into two groups, $$V_1 = \{0,...,n1-1\}$$ and $$V_2 = \{n1,...,n1+n2-1\}$$. Each $$u \in V_1$$ is connected to every $$v \in V_2$$.

INPUT:

• n1, n2 – number of vertices in each side
• set_position – boolean (default True); if set to True, we assign positions to the vertices so that the set of cardinality $$n1$$ is on the line $$y=1$$ and the set of cardinality $$n2$$ is on the line $$y=0$$.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each complete bipartite graph will be displayed with the first $$n1$$ nodes on the top row (at $$y=1$$) from left to right. The remaining $$n2$$ nodes appear at $$y=0$$, also from left to right. The shorter row (partition with fewer nodes) is stretched to the same length as the longer row, unless the shorter row has 1 node; in which case it is centered. The $$x$$ values in the plot are in domain $$[0, \max(n1, n2)]$$.

In the Complete Bipartite graph, there is a visual difference in using the spring-layout algorithm vs. the position dictionary used in this constructor. The position dictionary flattens the graph and separates the partitioned nodes, making it clear which nodes an edge is connected to. The Complete Bipartite graph plotted with the spring-layout algorithm tends to center the nodes in n1 (see spring_med in examples below), thus overlapping its nodes and edges, making it typically hard to decipher.

Filling the position dictionary in advance adds $$O(n)$$ to the constructor. Feel free to race the constructors below in the examples section. The much larger difference is the time added by the spring-layout algorithm when plotting. (Also shown in the example below). The spring model is typically described as $$O(n^3)$$, as appears to be the case in the NetworkX source code.

EXAMPLES:

Two ways of constructing the complete bipartite graph, using different layout algorithms:

sage: import networkx
sage: n = networkx.complete_bipartite_graph(389, 157); spring_big = Graph(n)   # long time
sage: posdict_big = graphs.CompleteBipartiteGraph(389, 157)                    # long time


Compare the plotting:

sage: n = networkx.complete_bipartite_graph(11, 17)
sage: spring_med = Graph(n)
sage: posdict_med = graphs.CompleteBipartiteGraph(11, 17)


Notice here how the spring-layout tends to center the nodes of $$n1$$:

sage: spring_med.show() # long time
sage: posdict_med.show() # long time


View many complete bipartite graphs with a Sage Graphics Array, with this constructor (i.e., the position dictionary filled):

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.CompleteBipartiteGraph(i+1,4)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


We compare to plotting with the spring-layout algorithm:

sage: g = []
sage: j = []
sage: for i in range(9):
....:     spr = networkx.complete_bipartite_graph(i+1,4)
....:     k = Graph(spr)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

sage: graphs.CompleteBipartiteGraph(5,6).complement()
complement(Complete bipartite graph of order 5+6): Graph on 11 vertices

static CompleteGraph(n)

Return a complete graph on n nodes.

A Complete Graph is a graph in which all nodes are connected to all other nodes.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each complete graph will be displayed with the first (0) node at the top, with the rest following in a counterclockwise manner.

In the complete graph, there is a big difference visually in using the spring-layout algorithm vs. the position dictionary used in this constructor. The position dictionary flattens the graph, making it clear which nodes an edge is connected to. But the complete graph offers a good example of how the spring-layout works. The edges push outward (everything is connected), causing the graph to appear as a 3-dimensional pointy ball. (See examples below).

EXAMPLES: We view many Complete graphs with a Sage Graphics Array, first with this constructor (i.e., the position dictionary filled):

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.CompleteGraph(i+3)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


We compare to plotting with the spring-layout algorithm:

sage: import networkx
sage: g = []
sage: j = []
sage: for i in range(9):
....:     spr = networkx.complete_graph(i+3)
....:     k = Graph(spr)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


Compare the constructors (results will vary)

sage: import networkx
sage: t = cputime()
sage: n = networkx.complete_graph(389); spring389 = Graph(n)
sage: cputime(t)           # random
0.59203700000000126
sage: t = cputime()
sage: posdict389 = graphs.CompleteGraph(389)
sage: cputime(t)           # random
0.6680419999999998


We compare plotting:

sage: import networkx
sage: n = networkx.complete_graph(23)
sage: spring23 = Graph(n)
sage: posdict23 = graphs.CompleteGraph(23)
sage: spring23.show() # long time
sage: posdict23.show() # long time

static CompleteMultipartiteGraph(l)

Returns a complete multipartite graph.

INPUT:

• l – a list of integers : the respective sizes of the components.

EXAMPLES:

A complete tripartite graph with sets of sizes $$5, 6, 8$$:

sage: g = graphs.CompleteMultipartiteGraph([5, 6, 8]); g
Multipartite Graph with set sizes [5, 6, 8]: Graph on 19 vertices


It clearly has a chromatic number of 3:

sage: g.chromatic_number()
3

static CossidentePenttilaGraph(q)

Cossidente-Penttila $$((q^3+1)(q+1)/2,(q^2+1)(q-1)/2,(q-3)/2,(q-1)^2/2)$$-strongly regular graph

For each odd prime power $$q$$, one can partition the points of the $$O_6^-(q)$$-generalized quadrange $$GQ(q,q^2)$$ into two parts, so that on any of them the induced subgraph of the point graph of the GQ has parameters as above [CP05].

Directly following the construction in [CP05] is not efficient, as one then needs to construct the dual $$GQ(q^2,q)$$. Thus we describe here a more efficient approach that we came up with, following a suggestion by T.Penttila. Namely, this partition is invariant under the subgroup $$H=\Omega_3(q^2)<O_6^-(q)$$. We build the appropriate $$H$$, which leaves the form $$B(X,Y,Z)=XY+Z^2$$ invariant, and pick up two orbits of $$H$$ on the $$F_q$$-points. One them is $$B$$-isotropic, and we take the representative $$(1:0:0)$$. The other one corresponds to the points of $$PG(2,q^2)$$ that have all the lines on them either missing the conic specified by $$B$$, or intersecting the conic in two points. We take $$(1:1:e)$$ as the representative. It suffices to pick $$e$$ so that $$e^2+1$$ is not a square in $$F_{q^2}$$. Indeed, The conic can be viewed as the union of $$\{(0:1:0)\}$$ and $$\{(1:-t^2:t) | t \in F_{q^2}\}$$. The coefficients of a generic line on $$(1:1:e)$$ are $$[1:-1-eb:b]$$, for $$-1\neq eb$$. Thus, to make sure the intersection with the conic is always even, we need that the discriminant of $$1+(1+eb)t^2+tb=0$$ never vanishes, and this is if and only if $$e^2+1$$ is not a square. Further, we need to adjust $$B$$, by multiplying it by appropriately chosen $$\nu$$, so that $$(1:1:e)$$ becomes isotropic under the relative trace norm $$\nu B(X,Y,Z)+(\nu B(X,Y,Z))^q$$. The latter is used then to define the graph.

INPUT:

• q – an odd prime power.

EXAMPLES:

For $$q=3$$ one gets Sims-Gewirtz graph.

sage: G=graphs.CossidentePenttilaGraph(3)    # optional - gap_packages (grape)
sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape)
(56, 10, 0, 2)


For $$q>3$$ one gets new graphs.

sage: G=graphs.CossidentePenttilaGraph(5)    # optional - gap_packages (grape)
sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape)
(378, 52, 1, 8)


REFERENCES:

 [CP05] (1, 2) A.Cossidente and T.Penttila Hemisystems on the Hermitian surface Journal of London Math. Soc. 72(2005), 731–741
static CoxeterGraph()

Return the Coxeter graph.

See the Wikipedia article Coxeter_graph.

EXAMPLES:

sage: g = graphs.CoxeterGraph()
sage: g.automorphism_group().cardinality()
336
sage: g.girth()
7
sage: g.chromatic_number()
3
sage: g.diameter()
4
sage: g.show(figsize=[10, 10]) # long time

static CubeGraph(n)

Returns the hypercube in $$n$$ dimensions.

The hypercube in $$n$$ dimension is build upon the binary strings on $$n$$ bits, two of them being adjacent if they differ in exactly one bit. Hence, the distance between two vertices in the hypercube is the Hamming distance.

EXAMPLES:

The distance between $$0100110$$ and $$1011010$$ is $$5$$, as expected

sage: g = graphs.CubeGraph(7)
sage: g.distance('0100110','1011010')
5


Plot several $$n$$-cubes in a Sage Graphics Array

sage: g = []
sage: j = []
sage: for i in range(6):
....:  k = graphs.CubeGraph(i+1)
....:  g.append(k)
...
sage: for i in range(2):
....:  n = []
....:  for m in range(3):
....:      n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:  j.append(n)
...
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show(figsize=[6,4]) # long time


Use the plot options to display larger $$n$$-cubes

sage: g = graphs.CubeGraph(9)
sage: g.show(figsize=[12,12],vertex_labels=False, vertex_size=20) # long time


AUTHORS:

• Robert Miller
static CycleGraph(n)

Return a cycle graph with n nodes.

A cycle graph is a basic structure which is also typically called an $$n$$-gon.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each cycle graph will be displayed with the first (0) node at the top, with the rest following in a counterclockwise manner.

The cycle graph is a good opportunity to compare efficiency of filling a position dictionary vs. using the spring-layout algorithm for plotting. Because the cycle graph is very symmetric, the resulting plots should be similar (in cases of small $$n$$).

Filling the position dictionary in advance adds $$O(n)$$ to the constructor.

EXAMPLES: Compare plotting using the predefined layout and networkx:

sage: import networkx
sage: n = networkx.cycle_graph(23)
sage: spring23 = Graph(n)
sage: posdict23 = graphs.CycleGraph(23)
sage: spring23.show() # long time
sage: posdict23.show() # long time


We next view many cycle graphs as a Sage graphics array. First we use the CycleGraph constructor, which fills in the position dictionary:

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.CycleGraph(i+3)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


Compare to plotting with the spring-layout algorithm:

sage: g = []
sage: j = []
sage: for i in range(9):
....:     spr = networkx.cycle_graph(i+3)
....:     k = Graph(spr)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static DegreeSequence(deg_sequence)

Returns a graph with the given degree sequence. Raises a NetworkX error if the proposed degree sequence cannot be that of a graph.

Graph returned is the one returned by the Havel-Hakimi algorithm, which constructs a simple graph by connecting vertices of highest degree to other vertices of highest degree, resorting the remaining vertices by degree and repeating the process. See Theorem 1.4 in [CharLes1996].

INPUT:

• deg_sequence - a list of integers with each entry corresponding to the degree of a different vertex.

EXAMPLES:

sage: G = graphs.DegreeSequence([3,3,3,3])
sage: G.edges(labels=False)
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
sage: G.show()  # long time

sage: G = graphs.DegreeSequence([3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3])
sage: G.show()  # long time

sage: G = graphs.DegreeSequence([4,4,4,4,4,4,4,4])
sage: G.show()  # long time

sage: G = graphs.DegreeSequence([1,2,3,4,3,4,3,2,3,2,1])
sage: G.show()  # long time


REFERENCE:

 [CharLes1996] Chartrand, G. and Lesniak, L.: Graphs and Digraphs. Chapman and Hall/CRC, 1996.
static DegreeSequenceBipartite(s1, s2)

Returns a bipartite graph whose two sets have the given degree sequences.

Given two different sequences of degrees $$s_1$$ and $$s_2$$, this functions returns ( if possible ) a bipartite graph on sets $$A$$ and $$B$$ such that the vertices in $$A$$ have $$s_1$$ as their degree sequence, while $$s_2$$ is the degree sequence of the vertices in $$B$$.

INPUT:

• s_1 – list of integers corresponding to the degree sequence of the first set.
• s_2 – list of integers corresponding to the degree sequence of the second set.

ALGORITHM:

This function works through the computation of the matrix given by the Gale-Ryser theorem, which is in this case the adjacency matrix of the bipartite graph.

EXAMPLES:

If we are given as sequences [2,2,2,2,2] and [5,5] we are given as expected the complete bipartite graph $$K_{2,5}$$

sage: g = graphs.DegreeSequenceBipartite([2,2,2,2,2],[5,5])
sage: g.is_isomorphic(graphs.CompleteBipartiteGraph(5,2))
True


Some sequences being incompatible if, for example, their sums are different, the functions raises a ValueError when no graph corresponding to the degree sequences exists.

sage: g = graphs.DegreeSequenceBipartite([2,2,2,2,1],[5,5])
Traceback (most recent call last):
...
ValueError: There exists no bipartite graph corresponding to the given degree sequences

static DegreeSequenceConfigurationModel(deg_sequence, seed=None)

Returns a random pseudograph with the given degree sequence. Raises a NetworkX error if the proposed degree sequence cannot be that of a graph with multiple edges and loops.

One requirement is that the sum of the degrees must be even, since every edge must be incident with two vertices.

INPUT:

• deg_sequence - a list of integers with each entry corresponding to the expected degree of a different vertex.
• seed - a random.Random seed or a Python int for the random number generator (default: None).

EXAMPLES:

sage: G = graphs.DegreeSequenceConfigurationModel([1,1])
[0 1]
[1 0]


Note: as of this writing, plotting of loops and multiple edges is not supported, and the output is allowed to contain both types of edges.

sage: G = graphs.DegreeSequenceConfigurationModel([3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3])
sage: len(G.edges())
30
sage: G.show()  # long time


REFERENCE:

 [Newman2003] Newman, M.E.J. The Structure and function of complex networks, SIAM Review vol. 45, no. 2 (2003), pp. 167-256.
static DegreeSequenceExpected(deg_sequence, seed=None)

Returns a random graph with expected given degree sequence. Raises a NetworkX error if the proposed degree sequence cannot be that of a graph.

One requirement is that the sum of the degrees must be even, since every edge must be incident with two vertices.

INPUT:

• deg_sequence - a list of integers with each entry corresponding to the expected degree of a different vertex.
• seed - a random.Random seed or a Python int for the random number generator (default: None).

EXAMPLES:

sage: G = graphs.DegreeSequenceExpected([1,2,3,2,3])
sage: G.edges(labels=False)
[(0, 3), (1, 3), (1, 4), (4, 4)]                    # 32-bit
[(0, 3), (1, 4), (2, 2), (2, 3), (2, 4), (4, 4)]    # 64-bit
sage: G.show()  # long time


REFERENCE:

 [ChungLu2002] Chung, Fan and Lu, L. Connected components in random graphs with given expected degree sequences. Ann. Combinatorics (6), 2002 pp. 125-145.
static DegreeSequenceTree(deg_sequence)

Returns a tree with the given degree sequence. Raises a NetworkX error if the proposed degree sequence cannot be that of a tree.

Since every tree has one more vertex than edge, the degree sequence must satisfy len(deg_sequence) - sum(deg_sequence)/2 == 1.

INPUT:

• deg_sequence - a list of integers with each entry corresponding to the expected degree of a different vertex.

EXAMPLES:

sage: G = graphs.DegreeSequenceTree([3,1,3,3,1,1,1,2,1])
sage: G.show()  # long time

static DejterGraph()

Return the Dejter graph.

The Dejter graph is obtained from the binary 7-cube by deleting a copy of the Hamming code of length 7. It is 6-regular, with 112 vertices and 336 edges. For more information, see the Wikipedia article Dejter_graph.

EXAMPLES:

sage: g = graphs.DejterGraph(); g
Dejter Graph: Graph on 112 vertices
sage: g.is_regular(k=6)
True
sage: g.girth()
4

static DesarguesGraph()

Return the Desargues graph.

PLOTTING: The layout chosen is the same as on the cover of [Har1994].

EXAMPLES:

sage: D = graphs.DesarguesGraph()
sage: L = graphs.LCFGraph(20,[5,-5,9,-9],5)
sage: D.is_isomorphic(L)
True
sage: D.show()  # long time

static DiamondGraph()

Returns a diamond graph with 4 nodes.

A diamond graph is a square with one pair of diagonal nodes connected.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the diamond graph is drawn as a diamond, with the first node on top, second on the left, third on the right, and fourth on the bottom; with the second and third node connected.

EXAMPLES: Construct and show a diamond graph

sage: g = graphs.DiamondGraph()
sage: g.show() # long time

static DipoleGraph(n)

Returns a dipole graph with n edges.

A dipole graph is a multigraph consisting of 2 vertices connected with n parallel edges.

EXAMPLES:

Construct and show a dipole graph with 13 edges:

sage: g = graphs.DipoleGraph(13); g
Dipole graph: Multi-graph on 2 vertices
sage: g.show() # long time

static DodecahedralGraph()

Returns a Dodecahedral graph (with 20 nodes)

The dodecahedral graph is cubic symmetric, so the spring-layout algorithm will be very effective for display. It is dual to the icosahedral graph.

PLOTTING: The Dodecahedral graph should be viewed in 3 dimensions. We chose to use the default spring-layout algorithm here, so that multiple iterations might yield a different point of reference for the user. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a string argument will be added to select the flat spring-layout over a future implementation.

EXAMPLES: Construct and show a Dodecahedral graph

sage: g = graphs.DodecahedralGraph()
sage: g.show() # long time


Create several dodecahedral graphs in a Sage graphics array They will be drawn differently due to the use of the spring-layout algorithm

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.DodecahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static DorogovtsevGoltsevMendesGraph(n)

Construct the n-th generation of the Dorogovtsev-Goltsev-Mendes graph.

EXAMPLES:

sage: G = graphs.DorogovtsevGoltsevMendesGraph(8)
sage: G.size()
6561


REFERENCE:

•  Dorogovtsev, S. N., Goltsev, A. V., and Mendes, J. F. F., Pseudofractal scale-free web, Phys. Rev. E 066122 (2002).
static DoubleStarSnark()

Return the double star snark.

The double star snark is a 3-regular graph on 30 vertices. See the Wikipedia article Double-star_snark.

EXAMPLES:

sage: g = graphs.DoubleStarSnark()
sage: g.order()
30
sage: g.size()
45
sage: g.chromatic_number()
3
sage: g.is_hamiltonian()
False
sage: g.automorphism_group().cardinality()
80
sage: g.show()

static DurerGraph()

Return the Dürer graph.

EXAMPLES:

The Dürer graph is named after Albrecht Dürer. It is a planar graph with 12 vertices and 18 edges.

sage: G = graphs.DurerGraph(); G
Durer graph: Graph on 12 vertices
sage: G.is_planar()
True
sage: G.order()
12
sage: G.size()
18


The Dürer graph has chromatic number 3, diameter 4, and girth 3.

sage: G.chromatic_number()
3
sage: G.diameter()
4
sage: G.girth()
3


Its automorphism group is isomorphic to $$D_6$$.

sage: ag = G.automorphism_group()
sage: ag.is_isomorphic(DihedralGroup(6))
True

static DyckGraph()

Return the Dyck graph.

For more information, see the MathWorld article on the Dyck graph or the Wikipedia article Dyck_graph.

EXAMPLES:

The Dyck graph was defined by Walther von Dyck in 1881. It has $$32$$ vertices and $$48$$ edges, and is a cubic graph (regular of degree $$3$$):

sage: G = graphs.DyckGraph(); G
Dyck graph: Graph on 32 vertices
sage: G.order()
32
sage: G.size()
48
sage: G.is_regular()
True
sage: G.is_regular(3)
True


It is non-planar and Hamiltonian, as well as bipartite (making it a bicubic graph):

sage: G.is_planar()
False
sage: G.is_hamiltonian()
True
sage: G.is_bipartite()
True


It has radius $$5$$, diameter $$5$$, and girth $$6$$:

sage: G.radius()
5
sage: G.diameter()
5
sage: G.girth()
6


Its chromatic number is $$2$$ and its automorphism group is of order $$192$$:

sage: G.chromatic_number()
2
sage: G.automorphism_group().cardinality()
192


It is a non-integral graph as it has irrational eigenvalues:

sage: G.characteristic_polynomial().factor()
(x - 3) * (x + 3) * (x - 1)^9 * (x + 1)^9 * (x^2 - 5)^6


It is a toroidal graph, and its embedding on a torus is dual to an embedding of the Shrikhande graph (ShrikhandeGraph).

static EgawaGraph(p, s)

Return the Egawa graph with parameters $$p$$, $$s$$.

Egawa graphs are a peculiar family of graphs devised by Yoshimi Egawa in [Ega1981] . The Shrikhande graph is a special case of this family of graphs, with parameters $$(1,0)$$. All the graphs in this family are not recognizable by 1-WL (Weisfeiler Lehamn algorithm of the first order) and 2-WL, that is their orbits are not correctly returned by k-WL for k lower than 3.

Furthermore, all the graphs in this family are distance-regular, but they are not distance-transitive if $$p \neq 0$$.

The Egawa graph with parameters $$(0, s)$$ is isomorphic to the Hamming graph with parameters $$(s, 4)$$, when the underlying set of the Hamming graph is $$[0,1,2,3]$$

INPUT:

• p – power to which the graph named $$Y$$ in the reference
provided above will be raised
• s – power to which the graph named $$X$$ in the reference
provided above will be raised

OUTPUT:

• G – The Egawa graph with parameters (p,s)

EXAMPLES:

Every Egawa graph is distance regular.

sage: g = graphs.EgawaGraph(1, 2)
sage: g.is_distance_regular()
True


An Egawa graph with parameters (0,s) is isomorphic to the Hamming graph with parameters (s, 4).

sage: g = graphs.EgawaGraph(0, 4)
sage: g.is_isomorphic(graphs.HammingGraph(4,4))
True

static EllinghamHorton54Graph()

Return the Ellingham-Horton 54-graph.

EXAMPLES:

This graph is 3-regular:

sage: g = graphs.EllinghamHorton54Graph()
sage: g.is_regular(k=3)
True


It is 3-connected and bipartite:

sage: g.vertex_connectivity() # not tested - too long
3
sage: g.is_bipartite()
True


It is not Hamiltonian:

sage: g.is_hamiltonian() # not tested - too long
False


… and it has a nice drawing

sage: g.show(figsize=[10, 10]) # not tested - too long

static EllinghamHorton78Graph()

Return the Ellingham-Horton 78-graph.

EXAMPLES:

This graph is 3-regular:

sage: g = graphs.EllinghamHorton78Graph()
sage: g.is_regular(k=3)
True


It is 3-connected and bipartite:

sage: g.vertex_connectivity() # not tested - too long
3
sage: g.is_bipartite()
True


It is not Hamiltonian:

sage: g.is_hamiltonian() # not tested - too long
False


… and it has a nice drawing

sage: g.show(figsize=[10,10]) # not tested - too long

static EmptyGraph()

Returns an empty graph (0 nodes and 0 edges).

This is useful for constructing graphs by adding edges and vertices individually or in a loop.

PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

EXAMPLES: Add one vertex to an empty graph and then show:

sage: empty1 = graphs.EmptyGraph()
0
sage: empty1.show() # long time


Use for loops to build a graph from an empty graph:

sage: empty2 = graphs.EmptyGraph()
sage: for i in range(5):
0
1
2
3
4
sage: for i in range(3):
sage: for i in range(1, 4):
sage: empty2.show() # long time

static ErreraGraph()

Return the Errera graph.

EXAMPLES:

The Errera graph is named after Alfred Errera. It is a planar graph on 17 vertices and having 45 edges.

sage: G = graphs.ErreraGraph(); G
Errera graph: Graph on 17 vertices
sage: G.is_planar()
True
sage: G.order()
17
sage: G.size()
45


The Errera graph is Hamiltonian with radius 3, diameter 4, girth 3, and chromatic number 4.

sage: G.is_hamiltonian()
True
3
sage: G.diameter()
4
sage: G.girth()
3
sage: G.chromatic_number()
4


Each vertex degree is either 5 or 6. That is, if $$f$$ counts the number of vertices of degree 5 and $$s$$ counts the number of vertices of degree 6, then $$f + s$$ is equal to the order of the Errera graph.

sage: D = G.degree_sequence()
sage: D.count(5) + D.count(6) == G.order()
True


The automorphism group of the Errera graph is isomorphic to the dihedral group of order 20.

sage: ag = G.automorphism_group()
sage: ag.is_isomorphic(DihedralGroup(10))
True

static EuropeMap(continental=False, year=2018)

Return European states as a graph of common border.

“European state” here is defined as an independent state having the capital city in Europe. The graph has an edge between those countries that have common land border.

INPUT:

• continental, a Boolean – if set, only return states in the continental Europe
• year – reserved for future use

EXAMPLES:

sage: Europe = graphs.EuropeMap(); Europe
Europe Map: Graph on 44 vertices
sage: Europe.neighbors('Ireland')
['United Kingdom']

sage: cont_Europe = graphs.EuropeMap(continental=True)
sage: cont_Europe.order()
40
sage: 'Iceland' in cont_Europe
False

static F26AGraph()

Return the F26A graph.

The F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges. For more information, see the Wikipedia article F26A_graph.

EXAMPLES:

sage: g = graphs.F26AGraph(); g
F26A Graph: Graph on 26 vertices
sage: g.order(),g.size()
(26, 39)
sage: g.automorphism_group().cardinality()
78
sage: g.girth()
6
sage: g.is_bipartite()
True
sage: g.characteristic_polynomial().factor()
(x - 3) * (x + 3) * (x^4 - 5*x^2 + 3)^6

static FibonacciTree(n)

Return the graph of the Fibonacci Tree $$F_{i}$$ of order $$n$$.

The Fibonacci tree $$F_{i}$$ is recursively defined as the tree with a root vertex and two attached child trees $$F_{i-1}$$ and $$F_{i-2}$$, where $$F_{1}$$ is just one vertex and $$F_{0}$$ is empty.

INPUT:

• n - the recursion depth of the Fibonacci Tree

EXAMPLES:

sage: g = graphs.FibonacciTree(3)
sage: g.is_tree()
True

sage: l1 = [ len(graphs.FibonacciTree(_)) + 1 for _ in range(6) ]
sage: l2 = list(fibonacci_sequence(2,8))
sage: l1 == l2
True


AUTHORS:

• Harald Schilly and Yann Laigle-Chapuy (2010-03-25)
static FlowerSnark()

Return a Flower Snark.

A flower snark has 20 vertices. It is part of the class of biconnected cubic graphs with edge chromatic number = 4, known as snarks. (i.e.: the Petersen graph). All snarks are not Hamiltonian, non-planar and have Petersen graph graph minors. See the Wikipedia article Flower_snark.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the nodes are drawn 0-14 on the outer circle, and 15-19 in an inner pentagon.

EXAMPLES: Inspect a flower snark:

sage: F = graphs.FlowerSnark()
sage: F
Flower Snark: Graph on 20 vertices
sage: F.graph6_string()
'[email protected][email protected][email protected]?Gp?K??C?CA?G?_G?Cc'


Now show it:

sage: F.show() # long time

static FoldedCubeGraph(n)

Returns the folded cube graph of order $$2^{n-1}$$.

The folded cube graph on $$2^{n-1}$$ vertices can be obtained from a cube graph on $$2^n$$ vertices by merging together opposed vertices. Alternatively, it can be obtained from a cube graph on $$2^{n-1}$$ vertices by adding an edge between opposed vertices. This second construction is the one produced by this method.

EXAMPLES:

The folded cube graph of order five is the Clebsch graph:

sage: fc = graphs.FoldedCubeGraph(5)
sage: clebsch = graphs.ClebschGraph()
sage: fc.is_isomorphic(clebsch)
True

static FolkmanGraph()

Return the Folkman graph.

See the Wikipedia article Folkman_graph.

EXAMPLES:

sage: g = graphs.FolkmanGraph()
sage: g.order()
20
sage: g.size()
40
sage: g.diameter()
4
sage: g.girth()
4
sage: g.charpoly().factor()
(x - 4) * (x + 4) * x^10 * (x^2 - 6)^4
sage: g.chromatic_number()
2
sage: g.is_eulerian()
True
sage: g.is_hamiltonian()
True
sage: g.is_vertex_transitive()
False
sage: g.is_bipartite()
True

static FosterGraph()

Return the Foster graph.

See the Wikipedia article Foster_graph.

EXAMPLES:

sage: g = graphs.FosterGraph()
sage: g.order()
90
sage: g.size()
135
sage: g.diameter()
8
sage: g.girth()
10
sage: g.automorphism_group().cardinality()
4320
sage: g.is_hamiltonian()
True

static FranklinGraph()

Return the Franklin graph.

EXAMPLES:

The Franklin graph is named after Philip Franklin. It is a 3-regular graph on 12 vertices and having 18 edges.

sage: G = graphs.FranklinGraph(); G
Franklin graph: Graph on 12 vertices
sage: G.is_regular(3)
True
sage: G.order()
12
sage: G.size()
18


The Franklin graph is a Hamiltonian, bipartite graph with radius 3, diameter 3, and girth 4.

sage: G.is_hamiltonian()
True
sage: G.is_bipartite()
True
3
sage: G.diameter()
3
sage: G.girth()
4


It is a perfect, triangle-free graph having chromatic number 2.

sage: G.is_perfect()
True
sage: G.is_triangle_free()
True
sage: G.chromatic_number()
2

static FriendshipGraph(n)

Return the friendship graph $$F_n$$.

The friendship graph is also known as the Dutch windmill graph. Let $$C_3$$ be the cycle graph on 3 vertices. Then $$F_n$$ is constructed by joining $$n \geq 1$$ copies of $$C_3$$ at a common vertex. If $$n = 1$$, then $$F_1$$ is isomorphic to $$C_3$$ (the triangle graph). If $$n = 2$$, then $$F_2$$ is the butterfly graph, otherwise known as the bowtie graph. For more information, see the Wikipedia article Friendship_graph.

INPUT:

• n – positive integer; the number of copies of $$C_3$$ to use in constructing $$F_n$$.

OUTPUT:

• The friendship graph $$F_n$$ obtained from $$n$$ copies of the cycle graph $$C_3$$.

EXAMPLES:

The first few friendship graphs.

sage: A = []; B = []
sage: for i in range(9):
....:     g = graphs.FriendshipGraph(i + 1)
....:     A.append(g)
sage: for i in range(3):
....:     n = []
....:     for j in range(3):
....:         n.append(A[3*i + j].plot(vertex_size=20, vertex_labels=False))
....:     B.append(n)
sage: G = sage.plot.graphics.GraphicsArray(B)
sage: G.show()  # long time


For $$n = 1$$, the friendship graph $$F_1$$ is isomorphic to the cycle graph $$C_3$$, whose visual representation is a triangle.

sage: G = graphs.FriendshipGraph(1); G
Friendship graph: Graph on 3 vertices
sage: G.show()  # long time
sage: G.is_isomorphic(graphs.CycleGraph(3))
True


For $$n = 2$$, the friendship graph $$F_2$$ is isomorphic to the butterfly graph, otherwise known as the bowtie graph.

sage: G = graphs.FriendshipGraph(2); G
Friendship graph: Graph on 5 vertices
sage: G.is_isomorphic(graphs.ButterflyGraph())
True


If $$n \geq 1$$, then the friendship graph $$F_n$$ has $$2n + 1$$ vertices and $$3n$$ edges. It has radius 1, diameter 2, girth 3, and chromatic number 3. Furthermore, $$F_n$$ is planar and Eulerian.

sage: n = randint(1, 10^3)
sage: G = graphs.FriendshipGraph(n)
sage: G.order() == 2*n + 1
True
sage: G.size() == 3*n
True
1
sage: G.diameter()
2
sage: G.girth()
3
sage: G.chromatic_number()
3
sage: G.is_planar()
True
sage: G.is_eulerian()
True

static FruchtGraph()

Return a Frucht Graph.

A Frucht graph has 12 nodes and 18 edges. It is the smallest cubic identity graph. It is planar and it is Hamiltonian. See the Wikipedia article Frucht_graph.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the first seven nodes are on the outer circle, with the next four on an inner circle and the last in the center.

EXAMPLES:

sage: FRUCHT = graphs.FruchtGraph()
sage: FRUCHT
Frucht graph: Graph on 12 vertices
sage: FRUCHT.graph6_string()
'KhCKM?_EGK?L'
sage: (graphs.FruchtGraph()).show() # long time

static FurerGadget(k, prefix=None)

Return a Furer gadget of order k and their coloring.

Construct the Furer gadget described in [CFI1992], a graph composed by a middle layer of $$2^(k-1)$$ nodes and two sets of nodes $$(a_0, ... , a_{k-1})$$ and $$(b_0, ... , b_{k-1})$$. Each node in the middle is connected to either $$a_i$$ or $$b_i$$, for each i in [0,k[. To read about the complete construction, see [CFI1992]. The returned coloring colors the middle section with one color, and then each pair $$(a_i, b_i)$$ with another color. Since this method is mainly used to create Furer gadgets for the Cai-Furer-Immerman construction, returning gadgets that don’t always have the same vertex labels is important, that’s why there is a parameter to manually set a prefix to be appended to each vertex label.

INPUT:

• k – The order of the returned Furer gadget, greater than 0.
• prefix – Prefix of to be appended to each vertex label,
so as to individualise the returned Furer gadget. Must be comparable for equality and hashable.

OUTPUT:

• G – The Furer gadget of order k
• coloring – A list of list of vertices, representing the
partition induced by the coloring of G’s vertices

EXAMPLES:

Furer gadget of order 3, without any prefix.

sage: G, p = graphs.FurerGadget(3)
sage: sorted(G, key=str)
[(), (0, 'a'), (0, 'b'), (0, 1), (0, 2),
(1, 'a'), (1, 'b'), (1, 2), (2, 'a'), (2, 'b')]
sage: sorted(G.edge_iterator(), key=str)
[((), (0, 'b'), None), ((), (1, 'b'), None),
((), (2, 'b'), None), ((0, 'b'), (1, 2), None),
((0, 1), (0, 'a'), None), ((0, 1), (1, 'a'), None),
((0, 1), (2, 'b'), None), ((0, 2), (0, 'a'), None),
((0, 2), (1, 'b'), None), ((0, 2), (2, 'a'), None),
((1, 2), (1, 'a'), None), ((1, 2), (2, 'a'), None)]


Furer gadget of order 3, with a prefix.

sage: G, p = graphs.FurerGadget(3, 'Prefix')
sage: sorted(G, key=str)
[('Prefix', ()), ('Prefix', (0, 'a')), ('Prefix', (0, 'b')),
('Prefix', (0, 1)), ('Prefix', (0, 2)), ('Prefix', (1, 'a')),
('Prefix', (1, 'b')), ('Prefix', (1, 2)), ('Prefix', (2, 'a')),
('Prefix', (2, 'b'))]
sage: sorted(G.edge_iterator(), key=str)
[(('Prefix', ()), ('Prefix', (0, 'b')), None),
(('Prefix', ()), ('Prefix', (1, 'b')), None),
(('Prefix', ()), ('Prefix', (2, 'b')), None),
(('Prefix', (0, 'b')), ('Prefix', (1, 2)), None),
(('Prefix', (0, 1)), ('Prefix', (0, 'a')), None),
(('Prefix', (0, 1)), ('Prefix', (1, 'a')), None),
(('Prefix', (0, 1)), ('Prefix', (2, 'b')), None),
(('Prefix', (0, 2)), ('Prefix', (0, 'a')), None),
(('Prefix', (0, 2)), ('Prefix', (1, 'b')), None),
(('Prefix', (0, 2)), ('Prefix', (2, 'a')), None),
(('Prefix', (1, 2)), ('Prefix', (1, 'a')), None),
(('Prefix', (1, 2)), ('Prefix', (2, 'a')), None)]

static FuzzyBallGraph(partition, q)

Construct a Fuzzy Ball graph with the integer partition partition and q extra vertices.

Let $$q$$ be an integer and let $$m_1,m_2,...,m_k$$ be a set of positive integers. Let $$n=q+m_1+...+m_k$$. The Fuzzy Ball graph with partition $$m_1,m_2,...,m_k$$ and $$q$$ extra vertices is the graph constructed from the graph $$G=K_n$$ by attaching, for each $$i=1,2,...,k$$, a new vertex $$a_i$$ to $$m_i$$ distinct vertices of $$G$$.

For given positive integers $$k$$ and $$m$$ and nonnegative integer $$q$$, the set of graphs FuzzyBallGraph(p, q) for all partitions $$p$$ of $$m$$ with $$k$$ parts are cospectral with respect to the normalized Laplacian.

EXAMPLES:

sage: F = graphs.FuzzyBallGraph([3,1],2)
[0 0 1 1 1 0 0 0]
[0 0 0 0 0 1 0 0]
[1 0 0 1 1 1 1 1]
[1 0 1 0 1 1 1 1]
[1 0 1 1 0 1 1 1]
[0 1 1 1 1 0 1 1]
[0 0 1 1 1 1 0 1]
[0 0 1 1 1 1 1 0]


Pick positive integers $$m$$ and $$k$$ and a nonnegative integer $$q$$. All the FuzzyBallGraphs constructed from partitions of $$m$$ with $$k$$ parts should be cospectral with respect to the normalized Laplacian:

sage: m=4; q=2; k=2
sage: g_list=[graphs.FuzzyBallGraph(p,q) for p in Partitions(m, length=k)]
sage: set([g.laplacian_matrix(normalized=True, vertices=list(g)).charpoly() for g in g_list])  # long time (7s on sage.math, 2011)
{x^8 - 8*x^7 + 4079/150*x^6 - 68689/1350*x^5 + 610783/10800*x^4 - 120877/3240*x^3 + 1351/100*x^2 - 931/450*x}

static GeneralizedPetersenGraph(n, k)

Returns a generalized Petersen graph with $$2n$$ nodes. The variables $$n$$, $$k$$ are integers such that $$n>2$$ and $$0<k\leq\lfloor(n-1)$$/$$2\rfloor$$

For $$k=1$$ the result is a graph isomorphic to the circular ladder graph with the same $$n$$. The regular Petersen Graph has $$n=5$$ and $$k=2$$. Other named graphs that can be described using this notation include the Desargues graph and the Möbius-Kantor graph.

INPUT:

• n - the number of nodes is $$2*n$$.
• k - integer $$0<k\leq\lfloor(n-1)$$/$$2\rfloor$$. Decides how inner vertices are connected.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the generalized Petersen graphs are displayed as an inner and outer cycle pair, with the first n nodes drawn on the outer circle. The first (0) node is drawn at the top of the outer-circle, moving counterclockwise after that. The inner circle is drawn with the (n)th node at the top, then counterclockwise as well.

EXAMPLES: For $$k=1$$ the resulting graph will be isomorphic to a circular ladder graph.

sage: g = graphs.GeneralizedPetersenGraph(13,1)
sage: g.is_isomorphic(g2)
True


The Desargues graph:

sage: g = graphs.GeneralizedPetersenGraph(10,3)
sage: g.girth()
6
sage: g.is_bipartite()
True


AUTHORS:

• Anders Jonsson (2009-10-15)
static GoethalsSeidelGraph(k, r)

Returns the graph $$\text{Goethals-Seidel}(k,r)$$.

The graph $$\text{Goethals-Seidel}(k,r)$$ comes from a construction presented in Theorem 2.4 of [GS70]. It relies on a (v,k)-BIBD with $$r$$ blocks and a hadamard_matrix() of order $$r+1$$. The result is a sage.graphs.strongly_regular_db.strongly_regular_graph() on $$v(r+1)$$ vertices with degree $$k=(n+r-1)/2$$.

It appears under this name in Andries Brouwer’s database of strongly regular graphs.

INPUT:

• k,r – integers

EXAMPLES:

sage: graphs.GoethalsSeidelGraph(3,3)
Graph on 28 vertices
sage: graphs.GoethalsSeidelGraph(3,3).is_strongly_regular(parameters=True)
(28, 15, 6, 10)

static GoldnerHararyGraph()

Return the Goldner-Harary graph.

EXAMPLES:

The Goldner-Harary graph is named after A. Goldner and Frank Harary. It is a planar graph having 11 vertices and 27 edges.

sage: G = graphs.GoldnerHararyGraph(); G
Goldner-Harary graph: Graph on 11 vertices
sage: G.is_planar()
True
sage: G.order()
11
sage: G.size()
27


The Goldner-Harary graph is chordal with radius 2, diameter 2, and girth 3.

sage: G.is_chordal()
True
2
sage: G.diameter()
2
sage: G.girth()
3


Its chromatic number is 4 and its automorphism group is isomorphic to the dihedral group $$D_6$$.

sage: G.chromatic_number()
4
sage: ag = G.automorphism_group()
sage: ag.is_isomorphic(DihedralGroup(6))
True

static GolombGraph()

Return the Golomb graph.

EXAMPLES:

The Golomb graph is a planar and Hamiltonian graph with 10 vertices and 18 edges. It has chromatic number 4, diameter 3, radius 2 and girth 3. It can be drawn in the plane as a unit distance graph:

sage: G = graphs.GolombGraph(); G
Golomb graph: Graph on 10 vertices
sage: pos = G.get_pos()
sage: dist2 = lambda u,v:(u-v)**2 + (u-v)**2
sage: all(dist2(pos[u], pos[v]) == 1 for u, v in G.edge_iterator(labels=None))
True

static GossetGraph()

Return the Gosset graph.

The Gosset graph is the skeleton of the Gosset_3_21() polytope. It has with 56 vertices and degree 27. For more information, see the Wikipedia article Gosset_graph.

EXAMPLES:

sage: g = graphs.GossetGraph(); g
Gosset Graph: Graph on 56 vertices

sage: g.order(), g.size()
(56, 756)

static GrayGraph(embedding=1)

Return the Gray graph.

See the Wikipedia article Gray_graph.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to 1 or 2.

EXAMPLES:

sage: g = graphs.GrayGraph()
sage: g.order()
54
sage: g.size()
81
sage: g.girth()
8
sage: g.diameter()
6
sage: g.show(figsize=[10, 10])   # long time
sage: graphs.GrayGraph(embedding = 2).show(figsize=[10, 10])   # long time

static Grid2dGraph(n1, n2, set_positions=True)

Returns a $$2$$-dimensional grid graph with $$n_1n_2$$ nodes ($$n_1$$ rows and $$n_2$$ columns).

A 2d grid graph resembles a $$2$$ dimensional grid. All inner nodes are connected to their $$4$$ neighbors. Outer (non-corner) nodes are connected to their $$3$$ neighbors. Corner nodes are connected to their 2 neighbors.

INPUT:

• n1 and n2 – two positive integers
• set_positions – (default: True) boolean use to prevent setting the position of the nodes.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, nodes are labelled in (row, column) pairs with $$(0, 0)$$ in the top left corner. Edges will always be horizontal and vertical - another advantage of filling the position dictionary.

EXAMPLES: Construct and show a grid 2d graph Rows = $$5$$, Columns = $$7$$

sage: g = graphs.Grid2dGraph(5,7)
sage: g.show() # long time

static GridGraph(dim_list)

Returns an n-dimensional grid graph.

INPUT:

• dim_list - a list of integers representing the number of nodes to extend in each dimension.

PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

EXAMPLES:

sage: G = graphs.GridGraph([2,3,4])
sage: G.show()  # long time

sage: C = graphs.CubeGraph(4)
sage: G = graphs.GridGraph([2,2,2,2])
sage: C.show()  # long time
sage: G.show()  # long time

static GrotzschGraph()

Return the Grötzsch graph.

The Grötzsch graph is an example of a triangle-free graph with chromatic number equal to 4. For more information, see the Wikipedia article Gr%C3%B6tzsch_graph.

EXAMPLES:

The Grötzsch graph is named after Herbert Grötzsch. It is a Hamiltonian graph with 11 vertices and 20 edges.

sage: G = graphs.GrotzschGraph(); G
Grotzsch graph: Graph on 11 vertices
sage: G.is_hamiltonian()
True
sage: G.order()
11
sage: G.size()
20


The Grötzsch graph is triangle-free and having radius 2, diameter 2, and girth 4.

sage: G.is_triangle_free()
True
2
sage: G.diameter()
2
sage: G.girth()
4


Its chromatic number is 4 and its automorphism group is isomorphic to the dihedral group $$D_5$$.

sage: G.chromatic_number()
4
sage: ag = G.automorphism_group()
sage: ag.is_isomorphic(DihedralGroup(5))
True

static HaemersGraph(q, hyperoval=None, hyperoval_matching=None, field=None, check_hyperoval=True)

Return the Haemers graph obtained from $$T_2^*(q)^*$$

Let $$q$$ be a power of 2. In Sect. 8.A of [BvL84] one finds a construction of a strongly regular graph with parameters $$(q^2(q+2),q^2+q-1,q-2,q)$$ from the graph of $$T_2^*(q)^*$$, constructed by T2starGeneralizedQuadrangleGraph(), by redefining adjacencies in the way specified by an arbitrary hyperoval_matching of the points (i.e. partitioning into size two parts) of hyperoval defining $$T_2^*(q)^*$$.

While [BvL84] gives the construction in geometric terms, it can be formulated, and is implemented, in graph-theoretic ones, of re-adjusting the edges. Namely, $$G=T_2^*(q)^*$$ has a partition into $$q+2$$ independent sets $$I_k$$ of size $$q^2$$ each. Each vertex in $$I_j$$ is adjacent to $$q$$ vertices from $$I_k$$. Each $$I_k$$ is paired to some $$I_{k'}$$, according to hyperoval_matching. One adds edges $$(s,t)$$ for $$s,t \in I_k$$ whenever $$s$$ and $$t$$ are adjacent to some $$u \in I_{k'}$$, and removes all the edges between $$I_k$$ and $$I_{k'}$$.

INPUT:

• q – a power of two
• hyperoval_matching – if None (default), pair each $$i$$-th point of hyperoval with $$(i+1)$$-th. Otherwise, specifies the pairing in the format $$((i_1,i'_1),(i_2,i'_2),...)$$.
• hyperoval – a hyperoval defining $$T_2^*(q)^*$$. If None (default), the classical hyperoval obtained from a conic is used. See the documentation of T2starGeneralizedQuadrangleGraph(), for more information.
• field – an instance of a finite field of order $$q$$, must be provided if hyperoval is provided.
• check_hyperoval – (default: True) if True, check hyperoval for correctness.

EXAMPLES:

using the built-in constructions:

sage: g=graphs.HaemersGraph(4); g
Haemers(4): Graph on 96 vertices
sage: g.is_strongly_regular(parameters=True)
(96, 19, 2, 4)


sage: g=graphs.HaemersGraph(4,hyperoval_matching=((0,5),(1,4),(2,3))); g
Haemers(4): Graph on 96 vertices
sage: g.is_strongly_regular(parameters=True)
(96, 19, 2, 4)

static HallJankoGraph(from_string=True)

Return the Hall-Janko graph.

For more information on the Hall-Janko graph, see the Wikipedia article Hall-Janko_graph.

The construction used to generate this graph in Sage is by a 100-point permutation representation of the Janko group $$J_2$$, as described in version 3 of the ATLAS of Finite Group representations, in particular on the page ATLAS: J2 – Permutation representation on 100 points.

INPUT:

• from_string (boolean) – whether to build the graph from its sparse6 string or through GAP. The two methods return the same graph though doing it through GAP takes more time. It is set to True by default.

EXAMPLES:

sage: g = graphs.HallJankoGraph()
sage: g.is_regular(36)
True
sage: g.is_vertex_transitive()
True


Is it really strongly regular with parameters 14, 12?

sage: nu = set(g.neighbors(0))
sage: for v in range(1, 100):
....:     if v in nu:
....:         expected = 14
....:     else:
....:         expected = 12
....:     nv = set(g.neighbors(v))
....:     if len(nu & nv) != expected:
....:         print("Something is wrong here!!!")
....:         break


Some other properties that we know how to check:

sage: g.diameter()
2
sage: g.girth()
3
sage: factor(g.characteristic_polynomial())
(x - 36) * (x - 6)^36 * (x + 4)^63

static HammingGraph(n, q, X=None)

Returns the Hamming graph with parameters n, q over X.

Hamming graphs are graphs over the cartesian product of n copies of X, where $$q = |X|$$, where the vertices, labelled with the corresponding tuple in $$X^n$$, are connected if the Hamming distance between their labels is 1. All Hamming graphs are regular, vertex-transitive and distance-regular.

Hamming graphs with parameters $$(1,q)$$ represent the complete graph with q vertices over the set X.

INPUT:

• n – power to which X will be raised to provide vertices
for the Hamming graph
• q – cardinality of X
• X – list of labels representing the vertices of the
underlying graph the Hamming graph will be based on; if None (or left unused), the list $$[0, ... , q-1]$$ will be used

OUTPUT:

• G – The Hamming graph with parameters $$(n,q,X)$$

EXAMPLES:

Every Hamming graph is distance-regular, regular and vertex-transitive.

sage: g = graphs.HammingGraph(3, 7)
sage: g.is_distance_regular()
True
sage: g.is_regular()
True
sage: g.is_vertex_transitive()
True


A Hamming graph with parameters (1,q) is isomorphic to the Complete graph with parameter q.

sage: g = graphs.HammingGraph(1, 23)
sage: g.is_isomorphic(graphs.CompleteGraph(23))
True


If a parameter q is provided which is not equal to X’s cardinality, an exception is raised.

sage: X = ['a','b','c','d','e']
sage: g = graphs.HammingGraph(2, 3, X)
Traceback (most recent call last):
...
ValueError: q must be the cardinality of X


REFERENCES:

For a more accurate description, see the following wikipedia page: Wikipedia article Hamming_graph

static HanoiTowerGraph(pegs, disks, labels=True, positions=True)

Returns the graph whose vertices are the states of the Tower of Hanoi puzzle, with edges representing legal moves between states.

INPUT:

• pegs - the number of pegs in the puzzle, 2 or greater
• disks - the number of disks in the puzzle, 1 or greater
• labels - default: True, if True the graph contains more meaningful labels, see explanation below. For large instances, turn off labels for much faster creation of the graph.
• positions - default: True, if True the graph contains layout information. This creates a planar layout for the case of three pegs. For large instances, turn off layout information for much faster creation of the graph.

OUTPUT:

The Tower of Hanoi puzzle has a certain number of identical pegs and a certain number of disks, each of a different radius. Initially the disks are all on a single peg, arranged in order of their radii, with the largest on the bottom.

The goal of the puzzle is to move the disks to any other peg, arranged in the same order. The one constraint is that the disks resident on any one peg must always be arranged with larger radii lower down.

The vertices of this graph represent all the possible states of this puzzle. Each state of the puzzle is a tuple with length equal to the number of disks, ordered by largest disk first. The entry of the tuple is the peg where that disk resides. Since disks on a given peg must go down in size as we go up the peg, this totally describes the state of the puzzle.

For example (2,0,0) means the large disk is on peg 2, the medium disk is on peg 0, and the small disk is on peg 0 (and we know the small disk must be above the medium disk). We encode these tuples as integers with a base equal to the number of pegs, and low-order digits to the right.

Two vertices are adjacent if we can change the puzzle from one state to the other by moving a single disk. For example, (2,0,0) is adjacent to (2,0,1) since we can move the small disk off peg 0 and onto (the empty) peg 1. So the solution to a 3-disk puzzle (with at least two pegs) can be expressed by the shortest path between (0,0,0) and (1,1,1). For more on this representation of the graph, or its properties, see [ARETT-DOREE].

For greatest speed we create graphs with integer vertices, where we encode the tuples as integers with a base equal to the number of pegs, and low-order digits to the right. So for example, in a 3-peg puzzle with 5 disks, the state (1,2,0,1,1) is encoded as $$1\ast 3^4 + 2\ast 3^3 + 0\ast 3^2 + 1\ast 3^1 + 1\ast 3^0 = 139$$.

For smaller graphs, the labels that are the tuples are informative, but slow down creation of the graph. Likewise computing layout information also incurs a significant speed penalty. For maximum speed, turn off labels and layout and decode the vertices explicitly as needed. The sage.rings.integer.Integer.digits() with the padsto option is a quick way to do this, though you may want to reverse the list that is output.

PLOTTING:

The layout computed when positions = True will look especially good for the three-peg case, when the graph is known to be planar. Except for two small cases on 4 pegs, the graph is otherwise not planar, and likely there is a better way to layout the vertices.

EXAMPLES:

A classic puzzle uses 3 pegs. We solve the 5 disk puzzle using integer labels and report the minimum number of moves required. Note that $$3^5-1$$ is the state where all 5 disks are on peg 2.

sage: H = graphs.HanoiTowerGraph(3, 5, labels=False, positions=False)
sage: H.distance(0, 3^5-1)
31


A slightly larger instance.

sage: H = graphs.HanoiTowerGraph(4, 6, labels=False, positions=False)
sage: H.num_verts()
4096
sage: H.distance(0, 4^6-1)
17


For a small graph, labels and layout information can be useful. Here we explicitly list a solution as a list of states.

sage: H = graphs.HanoiTowerGraph(3, 3, labels=True, positions=True)
sage: H.shortest_path((0,0,0), (1,1,1))
[(0, 0, 0), (0, 0, 1), (0, 2, 1), (0, 2, 2), (1, 2, 2), (1, 2, 0), (1, 1, 0), (1, 1, 1)]


Some facts about this graph with $$p$$ pegs and $$d$$ disks:

• only automorphisms are the “obvious” ones - renumber the pegs.
• chromatic number is less than or equal to $$p$$
• independence number is $$p^{d-1}$$
sage: H = graphs.HanoiTowerGraph(3,4,labels=False,positions=False)
sage: H.automorphism_group().is_isomorphic(SymmetricGroup(3))
True
sage: H.chromatic_number()
3
sage: len(H.independent_set()) == 3^(4-1)
True


Citations

 [ARETT-DOREE] Arett, Danielle and Doree, Suzanne “Coloring and counting on the Hanoi graphs” Mathematics Magazine, Volume 83, Number 3, June 2010, pages 200-9

AUTHOR:

• Rob Beezer, (2009-12-26), with assistance from Su Doree
static HararyGraph(k, n)

Returns the Harary graph on $$n$$ vertices and connectivity $$k$$, where $$2 \leq k < n$$.

A $$k$$-connected graph $$G$$ on $$n$$ vertices requires the minimum degree $$\delta(G)\geq k$$, so the minimum number of edges $$G$$ should have is $$\lceil kn/2\rceil$$. Harary graphs achieve this lower bound, that is, Harary graphs are minimal $$k$$-connected graphs on $$n$$ vertices.

The construction provided uses the method CirculantGraph. For more details, see the book D. B. West, Introduction to Graph Theory, 2nd Edition, Prentice Hall, 2001, p. 150–151; or the MathWorld article on Harary graphs.

EXAMPLES:

Harary graphs $$H_{k,n}$$:

sage: h = graphs.HararyGraph(5,9); h
Harary graph 5, 9: Graph on 9 vertices
sage: h.order()
9
sage: h.size()
23
sage: h.vertex_connectivity()
5

static HarborthGraph()

Return the Harborth Graph.

The Harborth graph has 104 edges and 52 vertices, and is the smallest known example of a 4-regular matchstick graph. For more information, see the Wikipedia article Harborth_graph.

EXAMPLES:

sage: g = graphs.HarborthGraph(); g
Harborth Graph: Graph on 52 vertices
sage: g.is_regular(4)
True

static HarriesGraph(embedding=1)

Return the Harries Graph.

The Harries graph is a Hamiltonian 3-regular graph on 70 vertices. See the Wikipedia article Harries_graph.

The default embedding here is to emphasize the graph’s 4 orbits. This graph actually has a funny construction. The following procedure gives an idea of it, though not all the adjacencies are being properly defined.

1. Take two disjoint copies of a Petersen graph. Their vertices will form an orbit of the final graph.
2. Subdivide all the edges once, to create 15+15=30 new vertices, which together form another orbit.
3. Create 15 vertices, each of them linked to 2 corresponding vertices of the previous orbit, one in each of the two subdivided Petersen graphs. At the end of this step all vertices from the previous orbit have degree 3, and the only vertices of degree 2 in the graph are those that were just created.
4. Create 5 vertices connected only to the ones from the previous orbit so that the graph becomes 3-regular.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to 1 or 2.

EXAMPLES:

sage: g = graphs.HarriesGraph()
sage: g.order()
70
sage: g.size()
105
sage: g.girth()
10
sage: g.diameter()
6
sage: g.show(figsize=[10, 10])   # long time
sage: graphs.HarriesGraph(embedding=2).show(figsize=[10, 10])   # long time

static HarriesWongGraph(embedding=1)

Return the Harries-Wong Graph.

The default embedding is an attempt to emphasize the graph’s 8 (!!!) different orbits. In order to understand this better, one can picture the graph as being built in the following way:

1. One first creates a 3-dimensional cube (8 vertices, 12 edges), whose vertices define the first orbit of the final graph.
2. The edges of this graph are subdivided once, to create 12 new vertices which define a second orbit.
3. The edges of the graph are subdivided once more, to create 24 new vertices giving a third orbit.
4. 4 vertices are created and made adjacent to the vertices of the second orbit so that they have degree 3. These 4 vertices also define a new orbit.
5. In order to make the vertices from the third orbit 3-regular (they all miss one edge), one creates a binary tree on 1 + 3 + 6 + 12 vertices. The leaves of this new tree are made adjacent to the 12 vertices of the third orbit, and the graph is now 3-regular. This binary tree contributes 4 new orbits to the Harries-Wong graph.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to 1 or 2.

EXAMPLES:

sage: g = graphs.HarriesWongGraph()
sage: g.order()
70
sage: g.size()
105
sage: g.girth()
10
sage: g.diameter()
6
sage: orbits = g.automorphism_group(orbits=True)[-1] # long time
sage: g.show(figsize=[15, 15], partition=orbits)     # long time


Alternative embedding:

sage: graphs.HarriesWongGraph(embedding=2).show()

static HeawoodGraph()

Return a Heawood graph.

The Heawood graph is a cage graph that has 14 nodes. It is a cubic symmetric graph. (See also the Möbius-Kantor graph). It is nonplanar and Hamiltonian. It has diameter = 3, radius = 3, girth = 6, chromatic number = 2. It is 4-transitive but not 5-transitive. See the Wikipedia article Heawood_graph.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the nodes are positioned in a circular layout with the first node appearing at the top, and then continuing counterclockwise.

EXAMPLES:

sage: H = graphs.HeawoodGraph()
sage: H
Heawood graph: Graph on 14 vertices
sage: H.graph6_string()
'[email protected][email protected]_G_'
sage: (graphs.HeawoodGraph()).show() # long time

static HerschelGraph()

Return the Herschel graph.

EXAMPLES:

The Herschel graph is named after Alexander Stewart Herschel. It is a planar, bipartite graph with 11 vertices and 18 edges.

sage: G = graphs.HerschelGraph(); G
Herschel graph: Graph on 11 vertices
sage: G.is_planar()
True
sage: G.is_bipartite()
True
sage: G.order()
11
sage: G.size()
18


The Herschel graph is a perfect graph with radius 3, diameter 4, and girth 4.

sage: G.is_perfect()
True
3
sage: G.diameter()
4
sage: G.girth()
4


Its chromatic number is 2 and its automorphism group is isomorphic to the dihedral group $$D_6$$.

sage: G.chromatic_number()
2
sage: ag = G.automorphism_group()
sage: ag.is_isomorphic(DihedralGroup(6))
True

static HexahedralGraph()

Returns a hexahedral graph (with 8 nodes).

A regular hexahedron is a 6-sided cube. The hexahedral graph corresponds to the connectivity of the vertices of the hexahedron. This graph is equivalent to a 3-cube.

PLOTTING: The hexahedral graph should be viewed in 3 dimensions. We chose to use the default spring-layout algorithm here, so that multiple iterations might yield a different point of reference for the user. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a string argument will be added to select the flat spring-layout over a future implementation.

EXAMPLES: Construct and show a Hexahedral graph

sage: g = graphs.HexahedralGraph()
sage: g.show() # long time


Create several hexahedral graphs in a Sage graphics array. They will be drawn differently due to the use of the spring-layout algorithm.

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.HexahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static HigmanSimsGraph(relabel=True)

Return the Higman-Sims graph.

The Higman-Sims graph is a remarkable strongly regular graph of degree 22 on 100 vertices. For example, it can be split into two sets of 50 vertices each, so that each half induces a subgraph isomorphic to the Hoffman-Singleton graph (HoffmanSingletonGraph()). This can be done in 352 ways (see Higman-Sims graph by Andries E. Brouwer, accessed 24 October 2009.)

Its most famous property is that the automorphism group has an index 2 subgroup which is one of the 26 sporadic groups. [HS1968]

The construction used here follows [Haf2004].

INPUT:

• relabel - default: True. If True the vertices will be labeled with consecutive integers. If False the labels are strings that are three digits long. “xyz” means the vertex is in group x (zero through three), pentagon or pentagram y (zero through four), and is vertex z (zero through four) of that pentagon or pentagram. See [Haf2004] for more.

OUTPUT:

The Higman-Sims graph.

EXAMPLES:

A split into the first 50 and last 50 vertices will induce two copies of the Hoffman-Singleton graph, and we illustrate another such split, which is obvious based on the construction used.

sage: H = graphs.HigmanSimsGraph()
sage: A = H.subgraph(range(0,50))
sage: B = H.subgraph(range(50,100))
sage: K = graphs.HoffmanSingletonGraph()
sage: K.is_isomorphic(A) and K.is_isomorphic(B)
True
sage: C = H.subgraph(range(25,75))
sage: D = H.subgraph(list(range(0,25))+list(range(75,100)))
sage: K.is_isomorphic(C) and K.is_isomorphic(D)
True


The automorphism group contains only one nontrivial proper normal subgroup, which is of index 2 and is simple. It is known as the Higman-Sims group.

sage: H = graphs.HigmanSimsGraph()
sage: G = H.automorphism_group()
sage: g=G.order(); g
88704000
sage: K = G.normal_subgroups()
sage: K.is_simple()
True
sage: g//K.order()
2


AUTHOR:

• Rob Beezer (2009-10-24)
static HoffmanGraph()

Return the Hoffman Graph.

See the Wikipedia article Hoffman_graph.

EXAMPLES:

sage: g = graphs.HoffmanGraph()
sage: g.is_bipartite()
True
sage: g.is_hamiltonian() # long time
True
3
sage: g.diameter()
4
sage: g.automorphism_group().cardinality()
48

static HoffmanSingletonGraph()

Return the Hoffman-Singleton graph.

The Hoffman-Singleton graph is the Moore graph of degree 7, diameter 2 and girth 5. The Hoffman-Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7 or 57. The first three respectively are the pentagon, the Petersen graph, and the Hoffman-Singleton graph. The existence of a Moore graph with girth 5 and degree 57 is still open.

A Moore graph is a graph with diameter $$d$$ and girth $$2d + 1$$. This implies that the graph is regular, and distance regular.

For more details, see [GR2001] and the Wikipedia article Hoffman–Singleton_graph.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. A novel algorithm written by Tom Boothby gives a random layout which is pleasing to the eye.

EXAMPLES:

sage: HS = graphs.HoffmanSingletonGraph()
sage: Set(HS.degree())
{7}
sage: HS.girth()
5
sage: HS.diameter()
2
sage: HS.num_verts()
50


Note that you get a different layout each time you create the graph.

sage: HS.layout()
(-0.844..., 0.535...)
sage: HS = graphs.HoffmanSingletonGraph()
sage: HS.layout()
(-0.904..., 0.425...)

static HoltGraph()

Return the Holt graph (also called the Doyle graph).

See the Wikipedia article Holt_graph.

EXAMPLES:

sage: g = graphs.HoltGraph();g
Holt graph: Graph on 27 vertices
sage: g.is_regular()
True
sage: g.is_vertex_transitive()
True
sage: g.chromatic_number()
3
sage: g.is_hamiltonian() # long time
True
3
sage: g.diameter()
3
sage: g.girth()
5
sage: g.automorphism_group().cardinality()
54

static HortonGraph()

Return the Horton Graph.

The Horton graph is a cubic 3-connected non-hamiltonian graph. For more information, see the Wikipedia article Horton_graph.

EXAMPLES:

sage: g = graphs.HortonGraph()
sage: g.order()
96
sage: g.size()
144
10
sage: g.diameter()
10
sage: g.girth()
6
sage: g.automorphism_group().cardinality()
96
sage: g.chromatic_number()
2
sage: g.is_hamiltonian() # not tested -- veeeery long
False

static HouseGraph()

Returns a house graph with 5 nodes.

A house graph is named for its shape. It is a triangle (roof) over a square (walls).

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the house graph is drawn with the first node in the lower-left corner of the house, the second in the lower-right corner of the house. The third node is in the upper-left corner connecting the roof to the wall, and the fourth is in the upper-right corner connecting the roof to the wall. The fifth node is the top of the roof, connected only to the third and fourth.

EXAMPLES: Construct and show a house graph

sage: g = graphs.HouseGraph()
sage: g.show() # long time

static HouseXGraph()

Returns a house X graph with 5 nodes.

A house X graph is a house graph with two additional edges. The upper-right corner is connected to the lower-left. And the upper-left corner is connected to the lower-right.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the house X graph is drawn with the first node in the lower-left corner of the house, the second in the lower-right corner of the house. The third node is in the upper-left corner connecting the roof to the wall, and the fourth is in the upper-right corner connecting the roof to the wall. The fifth node is the top of the roof, connected only to the third and fourth.

EXAMPLES: Construct and show a house X graph

sage: g = graphs.HouseXGraph()
sage: g.show() # long time

static HyperStarGraph(n, k)

Returns the hyper-star graph HS(n,k).

The vertices of the hyper-star graph are the set of binary strings of length n which contain k 1s. Two vertices, u and v, are adjacent only if u can be obtained from v by swapping the first bit with a different symbol in another position.

INPUT:

• n
• k

EXAMPLES:

sage: g = graphs.HyperStarGraph(6,3)
sage: g.plot() # long time
Graphics object consisting of 51 graphics primitives


REFERENCES:

• Lee, Hyeong-Ok, Jong-Seok Kim, Eunseuk Oh, and Hyeong-Seok Lim. “Hyper-Star Graph: A New Interconnection Network Improving the Network Cost of the Hypercube.” In Proceedings of the First EurAsian Conference on Information and Communication Technology, 858-865. Springer-Verlag, 2002.

AUTHORS:

• Michael Yurko (2009-09-01)
static IcosahedralGraph()

Returns an Icosahedral graph (with 12 nodes).

The regular icosahedron is a 20-sided triangular polyhedron. The icosahedral graph corresponds to the connectivity of the vertices of the icosahedron. It is dual to the dodecahedral graph. The icosahedron is symmetric, so the spring-layout algorithm will be very effective for display.

PLOTTING: The Icosahedral graph should be viewed in 3 dimensions. We chose to use the default spring-layout algorithm here, so that multiple iterations might yield a different point of reference for the user. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a string argument will be added to select the flat spring-layout over a future implementation.

EXAMPLES: Construct and show an Octahedral graph

sage: g = graphs.IcosahedralGraph()
sage: g.show() # long time


Create several icosahedral graphs in a Sage graphics array. They will be drawn differently due to the use of the spring-layout algorithm.

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.IcosahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static IntersectionGraph(S)

Returns the intersection graph of the family $$S$$

The intersection graph of a family $$S$$ is a graph $$G$$ with $$V(G)=S$$ such that two elements $$s_1,s_2\in S$$ are adjacent in $$G$$ if and only if $$s_1\cap s_2\neq \emptyset$$.

INPUT:

• S – a list of sets/tuples/iterables

Note

The elements of $$S$$ must be finite, hashable, and the elements of any $$s\in S$$ must be hashable too.

EXAMPLES:

sage: graphs.IntersectionGraph([(1,2,3),(3,4,5),(5,6,7)])
Intersection Graph: Graph on 3 vertices

static IntervalGraph(intervals, points_ordered=False)

Return the graph corresponding to the given intervals.

An interval graph is built from a list $$(a_i,b_i)_{1\leq i \leq n}$$ of intervals : to each interval of the list is associated one vertex, two vertices being adjacent if the two corresponding (closed) intervals intersect.

INPUT:

• intervals – the list of pairs $$(a_i,b_i)$$ defining the graph.
• points_ordered – states whether every interval $$(a_i,b_i)$$ of $$intervals$$ satisfies $$a_i<b_i$$. If satisfied then setting points_ordered to True will speed up the creation of the graph.

Note

• The vertices are named 0, 1, 2, and so on. The intervals used to create the graph are saved with the graph and can be recovered using get_vertex() or get_vertices().

EXAMPLES:

The following line creates the sequence of intervals $$(i, i+2)$$ for i in $$[0, ..., 8]$$:

sage: intervals = [(i,i+2) for i in range(9)]


In the corresponding graph

sage: g = graphs.IntervalGraph(intervals)
sage: g.get_vertex(3)
(3, 5)
sage: neigh = g.neighbors(3)
sage: for v in neigh: print(g.get_vertex(v))
(1, 3)
(2, 4)
(4, 6)
(5, 7)


The is_interval() method verifies that this graph is an interval graph.

sage: g.is_interval()
True


The intervals in the list need not be distinct.

sage: intervals = [ (1,2), (1,2), (1,2), (2,3), (3,4) ]
sage: g = graphs.IntervalGraph(intervals,True)
sage: g.clique_maximum()
[0, 1, 2, 3]
sage: g.get_vertices()
{0: (1, 2), 1: (1, 2), 2: (1, 2), 3: (2, 3), 4: (3, 4)}


The endpoints of the intervals are not ordered we get the same graph (except for the vertex labels).

sage: rev_intervals = [ (2,1), (2,1), (2,1), (3,2), (4,3) ]
sage: h = graphs.IntervalGraph(rev_intervals,False)
sage: h.get_vertices()
{0: (2, 1), 1: (2, 1), 2: (2, 1), 3: (3, 2), 4: (4, 3)}
sage: g.edges() == h.edges()
True

static IoninKharaghani765Graph()

Return a $$(765, 192, 48, 48)$$-strongly regular graph.

Existence of a strongly regular graph with these parameters was claimed in [IK2003]. Implementing the construction in the latter did not work, however. This function implements the following instructions, shared by Yury Ionin and Hadi Kharaghani.

Let $$A$$ be the affine plane over the field $$GF(3)=\{-1,0,1\}$$. Let

$\begin{split}\phi_1(x,y) &= x\\ \phi_2(x,y) &= y\\ \phi_3(x,y) &= x+y\\ \phi_4(x,y) &= x-y\\\end{split}$

For $$i=1,2,3,4$$ and $$j\in GF(3)$$, let $$L_{i,j}$$ be the line in $$A$$ defined by $$\phi_i(x,y)=j$$. Let $$\mathcal M$$ be the set of all 12 lines $$L_{i,j}$$, plus the empty set. Let $$\pi$$ be the permutation defined on $$\mathcal M$$ by $$\pi(L_{i,j}) = L_{i,j+1}$$ and $$\pi(\emptyset) = \emptyset$$, so that $$\pi$$ has three orbits of cardinality 3 and one of cardinality 1.

Let $$A=(p_1,...,p_9)$$ with $$p_1=(-1,1)$$, $$p_2=(-1,0)$$, $$p_3=(-1,1)$$, $$p_4=(0,-1)$$, $$p_5=(0,0)$$, $$p_6=(0,1)$$, $$p_7=(1,-1)$$, $$p_8=(1,0)$$, $$p_9=(1,1)$$. Note that $$p_i+p_{10-i}=(0,0)$$. For any subset $$X$$ of $$A$$, let $$M(X)$$ be the $$(0,1)$$-matrix of order 9 whose $$(i,j)$$-entry equals 1 if and only if $$p_{10-i}-p_j\in X$$. Note that $$M$$ is a symmetric matrix.

An $$MF$$-tuple is an ordered quintuple $$(X_1, X_2, X_3, X_4, X_5)$$ of subsets of $$A$$, of which one is the empty set and the other four are pairwise non-parallel lines. Such a quintuple generates the following block matrix:

$\begin{split}N(X_1, X_2, X_3, X_4, X_5) = \left( \begin{array}{ccccc} M(X_1) & M(X_2) & M(X_3) & M(X_4) & M(X_5)\\ M(X_2) & M(X_3) & M(X_4) & M(X_5) & M(X_1)\\ M(X_3) & M(X_4) & M(X_5) & M(X_1) & M(X_2)\\ M(X_4) & M(X_5) & M(X_1) & M(X_2) & M(X_3)\\ M(X_5) & M(X_1) & M(X_2) & M(X_3) & M(X_4) \end{array}\right)\end{split}$

Observe that if $$(X_1, X_2, X_3, X_4, X_5)$$ is an $$MF$$-tuple, then $$N(X_1, X_2, X_3, X_4, X_5)$$ is the symmetric incidence matrix of a symmetric $$(45, 12, 3)$$-design.

Let $$\mathcal F$$ be the set of all $$MF$$-tuples and let $$\sigma$$ be the following permutation of $$\mathcal F$$:

$\begin{split}\sigma(X_1, X_2, X_3, X_4, X_5) & = (X_2, X_3, X_4, X_5, X_1)\\ \pi(X_1, X_2, X_3, X_4, X_5) & = (\pi(X_1), \pi(X_2), \pi(X_3), \pi(X_4), \pi(X_5))\\\end{split}$

Observe that $$\sigma$$ and $$\pi$$ commute, and generate a (cyclic) group $$G$$ of order 15. We will from now on identify $$G$$ with the (cyclic) multiplicative group of the field $$GF(16)$$ equal to $$\{\omega^0,...,\omega^{14}\}$$. Let $$W=[w_{ij}]$$ be the following matrix of order 17 over $$GF(16)=\{a_1,...,a_16\}$$:

$\begin{split}w_{ij}=\left\{\begin{array}{ll} a_i+a_j & \text{if }1\leq i\leq 16, 1\leq j\leq 16,\\ 1 & \text{if }i=17, j\neq 17,\\ 1 & \text{if }i\neq 17, j= 17,\\ 0 & \text{if }i=j=17 \end{array}\right.\end{split}$

The diagonal entries of $$W$$ are equal to 0, each off-diagonal entry can be represented as $$\omega^k$$ with $$0\leq k\leq 14$$. Matrix $$W$$ is a symmetric $$BGW(17,16,15; G)$$.

Fix an $$MF$$-tuple $$(X_1, X_2, X_3, X_4, X_5)$$ and let $$S$$ be the block matrix obtained from $$W$$ by replacing every diagonal entry of $$W$$ by the zero matrix of order 45, and every off-diagonal entry $$\omega^k$$ by the matrix $$N(\sigma^k(X_1, X_2, X_3, X_4, X_5))$$ (through the association of $$\omega^k$$ with an element of $$G$$). Then $$S$$ is a symmetric incidence matrix of a symmetric $$(765, 192, 48)$$-design with zero diagonal, and therefore $$S$$ is an adjacency matrix of a strongly regular graph with parameters $$(765, 192, 48, 48)$$.

EXAMPLES:

sage: g = graphs.IoninKharaghani765Graph(); g
Ionin-Kharaghani: Graph on 765 vertices


Todo

An update to [IK2003] meant to fix the problem encountered became available 2016/02/24, see http://www.cs.uleth.ca/~hadi/research/IoninKharaghani.pdf

static JankoKharaghaniGraph(v)

Return a (936, 375, 150, 150)-srg or a (1800, 1029, 588, 588)-srg.

This functions returns a strongly regular graph for the two sets of parameters shown to be realizable in [JK2002]. The paper also uses a construction from [GM1987].

INPUT:

• v (integer) – one of 936 or 1800.

EXAMPLES:

sage: g = graphs.JankoKharaghaniGraph(936)   # long time
sage: g.is_strongly_regular(parameters=True) # long time
(936, 375, 150, 150)

sage: g = graphs.JankoKharaghaniGraph(1800)  # not tested (30s)
sage: g.is_strongly_regular(parameters=True) # not tested (30s)
(1800, 1029, 588, 588)

static JankoKharaghaniTonchevGraph()

Return a (324,153,72,72)-strongly regular graph from [JKT2001].

Build the graph using the description given in [JKT2001], taking sets B1 and B163 in the text as adjacencies of vertices 1 and 163, respectively, and taking the edge orbits of the group $$G$$ provided.

EXAMPLES:

sage: Gamma=graphs.JankoKharaghaniTonchevGraph()  # long time
sage: Gamma.is_strongly_regular(parameters=True)  # long time
(324, 153, 72, 72)

static JohnsonGraph(n, k)

Returns the Johnson graph with parameters $$n, k$$.

Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph $$J(n,k)$$ are the $$k$$-element subsets of an $$n$$-element set; two vertices are adjacent when they meet in a $$(k-1)$$-element set. See the Wikipedia article Johnson_graph for more information.

EXAMPLES:

The Johnson graph is a Hamiltonian graph.

sage: g = graphs.JohnsonGraph(7, 3)
sage: g.is_hamiltonian()
True


Every Johnson graph is vertex transitive.

sage: g = graphs.JohnsonGraph(6, 4)
sage: g.is_vertex_transitive()
True


The complement of the Johnson graph $$J(n,2)$$ is isomorphic to the Kneser Graph $$K(n,2)$$. In particular the complement of $$J(5,2)$$ is isomorphic to the Petersen graph.

sage: g = graphs.JohnsonGraph(5,2)
sage: g.complement().is_isomorphic(graphs.PetersenGraph())
True

static KingGraph(dim_list, radius=None, relabel=False)

Returns the $$d$$-dimensional King Graph with prescribed dimensions.

The 2-dimensional King Graph of parameters $$n$$ and $$m$$ is a graph with $$nm$$ vertices in which each vertex represents a square in an $$n \times m$$ chessboard, and each edge corresponds to a legal move by a king.

The d-dimensional King Graph with $$d >= 2$$ has for vertex set the cells of a d-dimensional grid with prescribed dimensions, and each edge corresponds to a legal move by a king in either one or two dimensions.

All 2-dimensional King Graphs are Hamiltonian, biconnected, and have chromatic number 4 as soon as both dimensions are larger or equal to 2.

INPUT:

• dim_list – an iterable object (list, set, dict) providing the dimensions $$n_1, n_2, \ldots, n_d$$, with $$n_i \geq 1$$, of the chessboard.
• radius – (default: None) by setting the radius to a positive integer, one may increase the power of the king to at least radius steps. When the radius equals the higher size of the dimensions, the resulting graph is a Queen Graph.
• relabel – (default: False) a boolean set to True if vertices must be relabeled as integers.

EXAMPLES:

The $$(2,2)$$-King Graph is isomorphic to the complete graph on 4 vertices:

sage: G = graphs.QueenGraph( [2, 2] )
sage: G.is_isomorphic( graphs.CompleteGraph(4) )
True


The King Graph with large enough radius is isomorphic to a Queen Graph:

sage: G = graphs.KingGraph( [5, 4], radius=5 )
sage: H = graphs.QueenGraph( [4, 5] )
sage: G.is_isomorphic( H )
True


Also True in higher dimensions:

sage: G = graphs.KingGraph( [2, 5, 4], radius=5 )
sage: H = graphs.QueenGraph( [4, 5, 2] )
sage: G.is_isomorphic( H )
True

static KittellGraph()

Return the Kittell Graph.

EXAMPLES:

sage: g = graphs.KittellGraph()
sage: g.order()
23
sage: g.size()
63
3
sage: g.diameter()
4
sage: g.girth()
3
sage: g.chromatic_number()
4

static Klein3RegularGraph()

Return the Klein 3-regular graph.

The cubic Klein graph has 56 vertices and can be embedded on a surface of genus 3. It is the dual of Klein7RegularGraph(). For more information, see the Wikipedia article Klein_graphs.

EXAMPLES:

sage: g = graphs.Klein3RegularGraph(); g
Klein 3-regular Graph: Graph on 56 vertices
sage: g.order(), g.size()
(56, 84)
sage: g.girth()
7
sage: g.automorphism_group().cardinality()
336
sage: g.chromatic_number()
3

static Klein7RegularGraph()

Return the Klein 7-regular graph.

The 7-valent Klein graph has 24 vertices and can be embedded on a surface of genus 3. It is the dual of Klein3RegularGraph(). For more information, see the Wikipedia article Klein_graphs.

EXAMPLES:

sage: g = graphs.Klein7RegularGraph(); g
Klein 7-regular Graph: Graph on 24 vertices
sage: g.order(), g.size()
(24, 84)
sage: g.girth()
3
sage: g.automorphism_group().cardinality()
336
sage: g.chromatic_number()
4

static KneserGraph(n, k)

Returns the Kneser Graph with parameters $$n, k$$.

The Kneser Graph with parameters $$n,k$$ is the graph whose vertices are the $$k$$-subsets of $$[0,1,\dots,n-1]$$, and such that two vertices are adjacent if their corresponding sets are disjoint.

For example, the Petersen Graph can be defined as the Kneser Graph with parameters $$5,2$$.

EXAMPLES:

sage: KG = graphs.KneserGraph(5,2)
sage: sorted(KG.vertex_iterator(), key=str)
[{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5},
{3, 4}, {3, 5}, {4, 5}]
sage: P = graphs.PetersenGraph()
sage: P.is_isomorphic(KG)
True

static KnightGraph(dim_list, one=1, two=2, relabel=False)

Returns the d-dimensional Knight Graph with prescribed dimensions.

The 2-dimensional Knight Graph of parameters $$n$$ and $$m$$ is a graph with $$nm$$ vertices in which each vertex represents a square in an $$n \times m$$ chessboard, and each edge corresponds to a legal move by a knight.

The d-dimensional Knight Graph with $$d >= 2$$ has for vertex set the cells of a d-dimensional grid with prescribed dimensions, and each edge corresponds to a legal move by a knight in any pairs of dimensions.

The $$(n,n)$$-Knight Graph is Hamiltonian for even $$n > 4$$.

INPUT:

• dim_list – an iterable object (list, set, dict) providing the dimensions $$n_1, n_2, \ldots, n_d$$, with $$n_i \geq 1$$, of the chessboard.
• one – (default: 1) integer indicating the number on steps in one dimension.
• two – (default: 2) integer indicating the number on steps in the second dimension.
• relabel – (default: False) a boolean set to True if vertices must be relabeled as integers.

EXAMPLES:

The $$(3,3)$$-Knight Graph has an isolated vertex:

sage: G = graphs.KnightGraph( [3, 3] )
sage: G.degree( (1,1) )
0


The $$(3,3)$$-Knight Graph minus vertex (1,1) is a cycle of order 8:

sage: G = graphs.KnightGraph( [3, 3] )
sage: G.delete_vertex( (1,1) )
sage: G.is_isomorphic( graphs.CycleGraph(8) )
True


The $$(6,6)$$-Knight Graph is Hamiltonian:

sage: G = graphs.KnightGraph( [6, 6] )
sage: G.is_hamiltonian()
True

static KrackhardtKiteGraph()

Return a Krackhardt kite graph with 10 nodes.

The Krackhardt kite graph was originally developed by David Krackhardt for the purpose of studying social networks (see [Kre2002] and the Wikipedia article Krackhardt_kite_graph). It is used to show the distinction between: degree centrality, betweeness centrality, and closeness centrality. For more information read the plotting section below in conjunction with the example.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the graph is drawn left to right, in top to bottom row sequence of [2, 3, 2, 1, 1, 1] nodes on each row. This places the fourth node (3) in the center of the kite, with the highest degree. But the fourth node only connects nodes that are otherwise connected, or those in its clique (i.e.: Degree Centrality). The eighth (7) node is where the kite meets the tail. It has degree = 3, less than the average, but is the only connection between the kite and tail (i.e.: Betweenness Centrality). The sixth and seventh nodes (5 and 6) are drawn in the third row and have degree = 5. These nodes have the shortest path to all other nodes in the graph (i.e.: Closeness Centrality). Please execute the example for visualization.

EXAMPLES:

Construct and show a Krackhardt kite graph

sage: g = graphs.KrackhardtKiteGraph()
sage: g.show() # long time

static LCFGraph(n, shift_list, repeats)

Return the cubic graph specified in LCF notation.

LCF (Lederberg-Coxeter-Fruchte) notation is a concise way of describing cubic Hamiltonian graphs. The way a graph is constructed is as follows. Since there is a Hamiltonian cycle, we first create a cycle on n nodes. The variable shift_list = [s_0, s_1, …, s_k-1] describes edges to be created by the following scheme: for each i, connect vertex i to vertex (i + s_i). Then, repeats specifies the number of times to repeat this process, where on the jth repeat we connect vertex (i + j*len(shift_list)) to vertex ( i + j*len(shift_list) + s_i).

INPUT:

• n - the number of nodes.
• shift_list - a list of integer shifts mod n.
• repeats - the number of times to repeat the process.

EXAMPLES:

sage: G = graphs.LCFGraph(4, [2,-2], 2)
sage: G.is_isomorphic(graphs.TetrahedralGraph())
True

sage: G = graphs.LCFGraph(20, [10,7,4,-4,-7,10,-4,7,-7,4], 2)
sage: G.is_isomorphic(graphs.DodecahedralGraph())
True

sage: G = graphs.LCFGraph(14, [5,-5], 7)
sage: G.is_isomorphic(graphs.HeawoodGraph())
True


The largest cubic nonplanar graph of diameter three:

sage: G = graphs.LCFGraph(20, [-10,-7,-5,4,7,-10,-7,-4,5,7,-10,-7,6,-5,7,-10,-7,5,-6,7], 1)
sage: G.degree()
[3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
sage: G.diameter()
3
sage: G.show()  # long time


PLOTTING: LCF Graphs are plotted as an n-cycle with edges in the middle, as described above.

REFERENCES:

•  Frucht, R. “A Canonical Representation of Trivalent Hamiltonian Graphs.” J. Graph Th. 1, 45-60, 1976.
•  Grunbaum, B. Convex Polytope es. New York: Wiley, pp. 362-364, 1967.
•  Lederberg, J. ‘DENDRAL-64: A System for Computer Construction, Enumeration and Notation of Organic Molecules as Tree Structures and Cyclic Graphs. Part II. Topology of Cyclic Graphs.’ Interim Report to the National Aeronautics and Space Administration. Grant NsG 81-60. December 15, 1965. http://profiles.nlm.nih.gov/BB/A/B/I/U/_/bbabiu.pdf.
static LadderGraph(n)

Returns a ladder graph with 2*n nodes.

A ladder graph is a basic structure that is typically displayed as a ladder, i.e.: two parallel path graphs connected at each corresponding node pair.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each ladder graph will be displayed horizontally, with the first n nodes displayed left to right on the top horizontal line.

EXAMPLES: Construct and show a ladder graph with 14 nodes

sage: g = graphs.LadderGraph(7)
sage: g.show() # long time


Create several ladder graphs in a Sage graphics array

sage: g = []
sage: j = []
sage: for i in range(9):
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static LivingstoneGraph()

Return the Livingstone Graph.

The Livingstone graph is a distance-transitive graph on 266 vertices whose automorphism group is the J1 group. For more information, see the Wikipedia article Livingstone_graph.

EXAMPLES:

sage: g = graphs.LivingstoneGraph() # optional - gap_packages internet
sage: g.order()                     # optional - gap_packages internet
266
sage: g.size()                      # optional - gap_packages internet
1463
sage: g.girth()                     # optional - gap_packages internet
5
sage: g.is_vertex_transitive()      # optional - gap_packages internet
True
sage: g.is_distance_regular()       # optional - gap_packages internet
True

static LjubljanaGraph(embedding=1)

Return the Ljubljana Graph.

The Ljubljana graph is a bipartite 3-regular graph on 112 vertices and 168 edges. It is not vertex-transitive as it has two orbits which are also independent sets of size 56. See the Wikipedia article Ljubljana_graph.

The default embedding is obtained from the Heawood graph.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to 1 or 2.

EXAMPLES:

sage: g = graphs.LjubljanaGraph()
sage: g.order()
112
sage: g.size()
168
sage: g.girth()
10
sage: g.diameter()
8
sage: g.show(figsize=[10, 10])   # long time
sage: graphs.LjubljanaGraph(embedding=2).show(figsize=[10, 10])   # long time

static LocalMcLaughlinGraph()

Return the local McLaughlin graph.

The local McLaughlin graph is a strongly regular graph with parameters $$(162,56,10,24)$$. It can be obtained from McLaughlinGraph() by considering the stabilizer of a point: one of its orbits has cardinality 162.

EXAMPLES:

sage: g = graphs.LocalMcLaughlinGraph(); g   # long time # optional - gap_packages
Local McLaughlin Graph: Graph on 162 vertices
sage: g.is_strongly_regular(parameters=True) # long time # optional - gap_packages
(162, 56, 10, 24)

static LollipopGraph(n1, n2)

Returns a lollipop graph with n1+n2 nodes.

A lollipop graph is a path graph (order n2) connected to a complete graph (order n1). (A barbell graph minus one of the bells).

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the complete graph will be drawn in the lower-left corner with the (n1)th node at a 45 degree angle above the right horizontal center of the complete graph, leading directly into the path graph.

EXAMPLES:

Construct and show a lollipop graph Candy = 13, Stick = 4:

sage: g = graphs.LollipopGraph(13,4); g
Lollipop graph: Graph on 17 vertices
sage: g.show() # long time

static M22Graph()

Return the M22 graph.

The $$M_{22}$$ graph is the unique strongly regular graph with parameters $$v = 77, k = 16, \lambda = 0, \mu = 4$$.

For more information on the $$M_{22}$$ graph, see https://www.win.tue.nl/~aeb/graphs/M22.html.

EXAMPLES:

sage: g = graphs.M22Graph()
sage: g.order()
77
sage: g.size()
616
sage: g.is_strongly_regular(parameters = True)
(77, 16, 0, 4)

static MarkstroemGraph()

Return the Markström Graph.

The Markström Graph is a cubic planar graph with no cycles of length 4 nor 8, but containing cycles of length 16. For more information, see the Wolfram page about the Markström Graph.

EXAMPLES:

sage: g = graphs.MarkstroemGraph()
sage: g.order()
24
sage: g.size()
36
sage: g.is_planar()
True
sage: g.is_regular(3)
True
sage: g.subgraph_search(graphs.CycleGraph(4)) is None
True
sage: g.subgraph_search(graphs.CycleGraph(8)) is None
True
sage: g.subgraph_search(graphs.CycleGraph(16))
Subgraph of (Markstroem Graph): Graph on 16 vertices

static MathonPseudocyclicMergingGraph(M, t)

Mathon’s merging of classes in a pseudo-cyclic 3-class association scheme

Construct strongly regular graphs from p.97 of [BvL84].

INPUT:

• M – the list of matrices in a pseudo-cyclic 3-class association scheme. The identity matrix must be the first entry.
• t (integer) – the number of the graph, from 0 to 2.

static MathonPseudocyclicStronglyRegularGraph(t, G=None, L=None)

Return a strongly regular graph on $$(4t+1)(4t-1)^2$$ vertices from [Mat78]

Let $$4t-1$$ be a prime power, and $$4t+1$$ be such that there exists a strongly regular graph $$G$$ with parameters $$(4t+1,2t,t-1,t)$$. In particular, $$4t+1$$ must be a sum of two squares [Mat78]. With this input, Mathon [Mat78] gives a construction of a strongly regular graph with parameters $$(4 \mu + 1, 2 \mu, \mu-1, \mu)$$, where $$\mu = t(4t(4t-1)-1)$$. The construction is optionally parametrised by an a skew-symmetric Latin square of order $$4t+1$$, with entries in $$-2t,...,-1,0,1,...,2t$$.

Our implementation follows a description given in [ST78].

INPUT:

• t – a positive integer
• G – if None (default), try to construct the necessary graph with parameters $$(4t+1,2t,t-1,t)$$, otherwise use the user-supplied one, with vertices labelled from $$0$$ to $$4t$$.
• L – if None (default), construct a necessary skew Latin square, otherwise use the user-supplied one. Here non-isomorphic Latin squares – one constructed from $$Z/9Z$$, and the other from $$(Z/3Z)^2$$ – lead to non-isomorphic graphs.

EXAMPLES:

Using default G and L.

sage: from sage.graphs.generators.families import MathonPseudocyclicStronglyRegularGraph
sage: G=MathonPseudocyclicStronglyRegularGraph(1); G
Mathon's PC SRG on 45 vertices: Graph on 45 vertices
sage: G.is_strongly_regular(parameters=True)
(45, 22, 10, 11)


Supplying G and L (constructed from the automorphism group of G).

sage: G = graphs.PaleyGraph(9)
sage: a = G.automorphism_group()
sage: it = (x for x in a.normal_subgroups() if x.order() == 9)
sage: subg = next(iter(it))
sage: r = [matrix(libgap.PermutationMat(libgap(z), 9).sage())
....:      for z in subg]
sage: ff = list(map(lambda y: (y-1,y-1),
....:          Permutation(map(lambda x: 1+r.index(x^-1), r)).cycle_tuples()[1:]))
sage: L = sum(i*(r[a]-r[b]) for i,(a,b) in zip(range(1,len(ff)+1), ff)); L
[ 0  1 -1 -3 -2 -4  3  4  2]
[-1  0  1 -4 -3 -2  2  3  4]
[ 1 -1  0 -2 -4 -3  4  2  3]
[ 3  4  2  0  1 -1 -3 -2 -4]
[ 2  3  4 -1  0  1 -4 -3 -2]
[ 4  2  3  1 -1  0 -2 -4 -3]
[-3 -2 -4  3  4  2  0  1 -1]
[-4 -3 -2  2  3  4 -1  0  1]
[-2 -4 -3  4  2  3  1 -1  0]

sage: G.relabel()
sage: G3x3=graphs.MathonPseudocyclicStronglyRegularGraph(2,G=G,L=L)
sage: G3x3.is_strongly_regular(parameters=True)
(441, 220, 109, 110)
sage: G3x3.automorphism_group(algorithm="bliss").order() # optional - bliss
27
sage: G9=graphs.MathonPseudocyclicStronglyRegularGraph(2)
sage: G9.is_strongly_regular(parameters=True)
(441, 220, 109, 110)
sage: G9.automorphism_group(algorithm="bliss").order() # optional - bliss
9


REFERENCES:

 [Mat78] (1, 2, 3) R. A. Mathon, Symmetric conference matrices of order $$pq^2 + 1$$, Canad. J. Math. 30 (1978) 321-331
 [ST78] J. J. Seidel and D. E. Taylor, Two-graphs, a second survey. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978), pp. 689–711, Colloq. Math. Soc. János Bolyai, 25, North-Holland, Amsterdam-New York, 1981.
static MathonStronglyRegularGraph(t)

Return one of Mathon’s graphs on 784 vertices.

INPUT:

• t (integer) – the number of the graph, from 0 to 2.

EXAMPLES:

sage: from sage.graphs.generators.smallgraphs import MathonStronglyRegularGraph
sage: G = MathonStronglyRegularGraph(0)        # long time
sage: G.is_strongly_regular(parameters=True)   # long time
(784, 243, 82, 72)

static McGeeGraph(embedding=2)

Return the McGee Graph.

See the Wikipedia article McGee_graph.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to 1 or 2.

EXAMPLES:

sage: g = graphs.McGeeGraph()
sage: g.order()
24
sage: g.size()
36
sage: g.girth()
7
sage: g.diameter()
4
sage: g.show()
sage: graphs.McGeeGraph(embedding=1).show()

static McLaughlinGraph()

Return the McLaughlin Graph.

The McLaughlin Graph is the unique strongly regular graph of parameters $$(275, 112, 30, 56)$$.

For more information on the McLaughlin Graph, see its web page on Andries Brouwer’s website which gives the definition that this method implements.

Note

To create this graph you must have the gap_packages spkg installed.

EXAMPLES:

sage: g = graphs.McLaughlinGraph()           # optional gap_packages
sage: g.is_strongly_regular(parameters=True) # optional gap_packages
(275, 112, 30, 56)
sage: set(g.spectrum()) == {112, 2, -28}     # optional gap_packages
True

static MeredithGraph()

Return the Meredith Graph.

The Meredith Graph is a 4-regular 4-connected non-hamiltonian graph. For more information on the Meredith Graph, see the Wikipedia article Meredith_graph.

EXAMPLES:

sage: g = graphs.MeredithGraph()
sage: g.is_regular(4)
True
sage: g.order()
70
sage: g.size()
140
7
sage: g.diameter()
8
sage: g.girth()
4
sage: g.chromatic_number()
3
sage: g.is_hamiltonian() # long time
False

static MoebiusKantorGraph()

Return a Möbius-Kantor Graph.

A Möbius-Kantor graph is a cubic symmetric graph. (See also the Heawood graph). It has 16 nodes and 24 edges. It is nonplanar and Hamiltonian. It has diameter = 4, girth = 6, and chromatic number = 2. It is identical to the Generalized Petersen graph, P[8,3].

For more details, see Möbius-Kantor Graph - from Wolfram MathWorld.

PLOTTING: See the plotting section for the generalized Petersen graphs.

EXAMPLES:

sage: MK = graphs.MoebiusKantorGraph()
sage: MK
Moebius-Kantor Graph: Graph on 16 vertices
sage: MK.graph6_string()
'[email protected][email protected]?Q_AS'
sage: (graphs.MoebiusKantorGraph()).show() # long time

static MoserSpindle()

Return the Moser spindle.

EXAMPLES:

The Moser spindle is a planar graph having 7 vertices and 11 edges:

sage: G = graphs.MoserSpindle(); G
Moser spindle: Graph on 7 vertices
sage: G.is_planar()
True
sage: G.order()
7
sage: G.size()
11


It is a Hamiltonian graph with radius 2, diameter 2, and girth 3:

sage: G.is_hamiltonian()
True
2
sage: G.diameter()
2
sage: G.girth()
3


The Moser spindle can be drawn in the plane as a unit distance graph, has chromatic number 4, and its automorphism group is isomorphic to the dihedral group $$D_4$$:

sage: pos = G.get_pos()
sage: all(sum((ui-vi)**2 for ui, vi in zip(pos[u], pos[v])) == 1
....:         for u, v in G.edge_iterator(labels=None))
True
sage: G.chromatic_number()
4
sage: ag = G.automorphism_group()
sage: ag.is_isomorphic(DihedralGroup(4))
True

static MuzychukS6Graph(n, d, Phi='fixed', Sigma='fixed', verbose=False)

Return a strongly regular graph of S6 type from [Mu07] on $$n^d((n^d-1)/(n-1)+1)$$ vertices

The construction depends upon a number of parameters, two of them, $$n$$ and $$d$$, mandatory, and $$\Phi$$ and $$\Sigma$$ mappings defined in [Mu07]. These graphs have parameters $$(mn^d, n^{d-1}(m-1) - 1,\mu - 2,\mu)$$, where $$\mu=\frac{n^{d-1}-1}{n-1}n^{d-1}$$ and $$m:=\frac{n^d-1}{n-1}+1$$.

Some details on $$\Phi$$ and $$\Sigma$$ are as follows. Let $$L$$ be the complete graph on $$M:=\{0,..., m-1\}$$ with the matching $$\{(2i,2i+1) | i=0,...,m/2\}$$ removed. Then one arbitrarily chooses injections $$\Phi_i$$ from the edges of $$L$$ on $$i \in M$$ into sets of parallel classes of affine $$d$$-dimensional designs; our implementation uses the designs of hyperplanes in $$d$$-dimensional affine geometries over $$GF(n)$$. Finally, for each edge $$ij$$ of $$L$$ one arbitrarily chooses bijections $$\Sigma_{ij}$$ between $$\Phi_i$$ and $$\Phi_j$$. More details, in particular how these choices lead to non-isomorphic graphs, are in [Mu07].

INPUT:

• n (integer)– a prime power

• d (integer)– must be odd if $$n$$ is odd

• Phi is an optional parameter of the construction; it must be either

• ‘fixed’– this will generate fixed default $$\Phi_i$$, for $$i \in M$$, or
• ‘random’– $$\Phi_i$$ are generated at random, or
• A dictionary describing the functions $$\Phi_i$$; for $$i \in M$$, Phi[(i, T)] in $$M$$, for each edge T of $$L$$ on $$i$$. Also, each $$\Phi_i$$ must be injective.
• Sigma is an optional parameter of the construction; it must be either

• ‘fixed’– this will generate a fixed default $$\Sigma$$, or
• ‘random’– $$\Sigma$$ is generated at random.
• verbose (Boolean)– default is False. If True, print progress information

Todo

Implement the possibility to explicitly supply the parameter $$\Sigma$$ of the construction.

EXAMPLES:

sage: graphs.MuzychukS6Graph(3, 3).is_strongly_regular(parameters=True)
(378, 116, 34, 36)
sage: phi={(2,(0,2)):0,(1,(1,3)):1,(0,(0,3)):1,(2,(1,2)):1,(1,(1,
....:  2)):0,(0,(0,2)):0,(3,(0,3)):0,(3,(1,3)):1}
sage: graphs.MuzychukS6Graph(2,2,Phi=phi).is_strongly_regular(parameters=True)
(16, 5, 0, 2)


REFERENCE:

 [Mu07] (1, 2, 3) M. Muzychuk. A generalization of Wallis-Fon-Der-Flaass construction of strongly regular graphs. J. Algebraic Combin., 25(2):169–187, 2007.
static MycielskiGraph(k=1, relabel=True)

Returns the $$k$$-th Mycielski Graph.

The graph $$M_k$$ is triangle-free and has chromatic number equal to $$k$$. These graphs show, constructively, that there are triangle-free graphs with arbitrarily high chromatic number.

The Mycielski graphs are built recursively starting with $$M_0$$, an empty graph; $$M_1$$, a single vertex graph; and $$M_2$$ is the graph $$K_2$$. $$M_{k+1}$$ is then built from $$M_k$$ as follows:

If the vertices of $$M_k$$ are $$v_1,\ldots,v_n$$, then the vertices of $$M_{k+1}$$ are $$v_1,\ldots,v_n,w_1,\ldots,w_n,z$$. Vertices $$v_1,\ldots,v_n$$ induce a copy of $$M_k$$. Vertices $$w_1,\ldots,w_n$$ are an independent set. Vertex $$z$$ is adjacent to all the $$w_i$$-vertices. Finally, vertex $$w_i$$ is adjacent to vertex $$v_j$$ iff $$v_i$$ is adjacent to $$v_j$$.

INPUT:

• k Number of steps in the construction process.
• relabel Relabel the vertices so their names are the integers range(n) where n is the number of vertices in the graph.

EXAMPLES:

The Mycielski graph $$M_k$$ is triangle-free and has chromatic number equal to $$k$$.

sage: g = graphs.MycielskiGraph(5)
sage: g.is_triangle_free()
True
sage: g.chromatic_number()
5


The graphs $$M_4$$ is (isomorphic to) the Grotzsch graph.

sage: g = graphs.MycielskiGraph(4)
sage: g.is_isomorphic(graphs.GrotzschGraph())
True


REFERENCES:

static MycielskiStep(g)

Perform one iteration of the Mycielski construction.

See the documentation for MycielskiGraph which uses this method. We expose it to all users in case they may find it useful.

EXAMPLE. One iteration of the Mycielski step applied to the 5-cycle yields a graph isomorphic to the Grotzsch graph

sage: g = graphs.CycleGraph(5)
sage: h = graphs.MycielskiStep(g)
sage: h.is_isomorphic(graphs.GrotzschGraph())
True

static NKStarGraph(n, k)

Returns the (n,k)-star graph.

The vertices of the (n,k)-star graph are the set of all arrangements of n symbols into labels of length k. There are two adjacency rules for the (n,k)-star graph. First, two vertices are adjacent if one can be obtained from the other by swapping the first symbol with another symbol. Second, two vertices are adjacent if one can be obtained from the other by swapping the first symbol with an external symbol (a symbol not used in the original label).

INPUT:

• n
• k

EXAMPLES:

sage: g = graphs.NKStarGraph(4,2)
sage: g.plot() # long time
Graphics object consisting of 31 graphics primitives


REFERENCES:

• Wei-Kuo, Chiang, and Chen Rong-Jaye. “The (n, k)-star graph: A generalized star graph.” Information Processing Letters 56, no. 5 (December 8, 1995): 259-264.

AUTHORS:

• Michael Yurko (2009-09-01)
static NStarGraph(n)

Returns the n-star graph.

The vertices of the n-star graph are the set of permutations on n symbols. There is an edge between two vertices if their labels differ only in the first and one other position.

INPUT:

• n

EXAMPLES:

sage: g = graphs.NStarGraph(4)
sage: g.plot() # long time
Graphics object consisting of 61 graphics primitives


REFERENCES:

• S.B. Akers, D. Horel and B. Krishnamurthy, The star graph: An attractive alternative to the previous n-cube. In: Proc. Internat. Conf. on Parallel Processing (1987), pp. 393–400.

AUTHORS:

• Michael Yurko (2009-09-01)
static NauruGraph(embedding=2)

Return the Nauru Graph.

See the Wikipedia article Nauru_graph.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to 1 or 2.

EXAMPLES:

sage: g = graphs.NauruGraph()
sage: g.order()
24
sage: g.size()
36
sage: g.girth()
6
sage: g.diameter()
4
sage: g.show()
sage: graphs.NauruGraph(embedding=1).show()

static NonisotropicOrthogonalPolarGraph(m, q, sign='+', perp=None)

Returns the Graph $$NO^{\epsilon,\perp}_{m}(q)$$

Let the vectorspace of dimension $$m$$ over $$F_q$$ be endowed with a nondegenerate quadratic form $$F$$, of type sign for $$m$$ even.

• $$m$$ even: assume further that $$q=2$$ or $$3$$. Returns the graph of the points (in the underlying projective space) $$x$$ satisfying $$F(x)=1$$, with adjacency given by orthogonality w.r.t. $$F$$. Parameter perp is ignored.
• $$m$$ odd: if perp is not None, then we assume that $$q=5$$ and return the graph of the points $$x$$ satisfying $$F(x)=\pm 1$$ if sign="+", respectively $$F(x) \in \{2,3\}$$ if sign="-", with adjacency given by orthogonality w.r.t. $$F$$ (cf. Sect 7.D of [BvL84]). Otherwise return the graph of nongenerate hyperplanes of type sign, adjacent whenever the intersection is degenerate (cf. Sect. 7.C of [BvL84]). Note that for $$q=2$$ one will get a complete graph.

For more information, see Sect. 9.9 of [BH12] and [BvL84]. Note that the page of Andries Brouwer’s website uses different notation.

INPUT:

• m - integer, half the dimension of the underlying vectorspace
• q - a power of a prime number, the size of the underlying field
• sign"+" (default) or "-".

EXAMPLES:

$$NO^-(4,2)$$ is isomorphic to Petersen graph:

sage: g=graphs.NonisotropicOrthogonalPolarGraph(4,2,'-'); g
NO^-(4, 2): Graph on 10 vertices
sage: g.is_strongly_regular(parameters=True)
(10, 3, 0, 1)


$$NO^-(6,2)$$ and $$NO^+(6,2)$$:

sage: g=graphs.NonisotropicOrthogonalPolarGraph(6,2,'-')
sage: g.is_strongly_regular(parameters=True)
(36, 15, 6, 6)
sage: g=graphs.NonisotropicOrthogonalPolarGraph(6,2,'+'); g
NO^+(6, 2): Graph on 28 vertices
sage: g.is_strongly_regular(parameters=True)
(28, 15, 6, 10)


$$NO^+(8,2)$$:

sage: g=graphs.NonisotropicOrthogonalPolarGraph(8,2,'+')
sage: g.is_strongly_regular(parameters=True)
(120, 63, 30, 36)


Wilbrink’s graphs for $$q=5$$:

sage: graphs.NonisotropicOrthogonalPolarGraph(5,5,perp=1).is_strongly_regular(parameters=True) # long time
(325, 60, 15, 10)
sage: graphs.NonisotropicOrthogonalPolarGraph(5,5,'-',perp=1).is_strongly_regular(parameters=True) # long time
(300, 65, 10, 15)


Wilbrink’s graphs:

sage: g=graphs.NonisotropicOrthogonalPolarGraph(5,4,'+')
sage: g.is_strongly_regular(parameters=True)
(136, 75, 42, 40)
sage: g=graphs.NonisotropicOrthogonalPolarGraph(5,4,'-')
sage: g.is_strongly_regular(parameters=True)
(120, 51, 18, 24)
sage: g=graphs.NonisotropicOrthogonalPolarGraph(7,4,'+'); g # not tested (long time)
NO^+(7, 4): Graph on 2080 vertices
sage: g.is_strongly_regular(parameters=True) # not tested (long time)
(2080, 1071, 558, 544)

static NonisotropicUnitaryPolarGraph(m, q)

Returns the Graph $$NU(m,q)$$.

Returns the graph on nonisotropic, with respect to a nondegenerate Hermitean form, points of the $$(m-1)$$-dimensional projective space over $$F_q$$, with points adjacent whenever they lie on a tangent (to the set of isotropic points) line. For more information, see Sect. 9.9 of [BH12] and series C14 in [Hu75].

INPUT:

• m,q (integers) – $$q$$ must be a prime power.

EXAMPLES:

sage: g=graphs.NonisotropicUnitaryPolarGraph(5,2); g
NU(5, 2): Graph on 176 vertices
sage: g.is_strongly_regular(parameters=True)
(176, 135, 102, 108)


REFERENCE:

 [Hu75] (1, 2) X. L. Hubaut. Strongly regular graphs. Disc. Math. 13(1975), pp 357–381. doi:10.1016/0012-365X(75)90057-6
static Nowhere0WordsTwoWeightCodeGraph(q, hyperoval=None, field=None, check_hyperoval=True)

Return the subgraph of nowhere 0 words from two-weight code of projective plane hyperoval.

Let $$q=2^k$$ and $$\Pi=PG(2,q)$$. Fix a hyperoval $$O \subset \Pi$$. Let $$V=F_q^3$$ and $$C$$ the two-weight 3-dimensional linear code over $$F_q$$ with words $$c(v)$$ obtained from $$v\in V$$ by computing

$c(v)=(\langle v,o_1 \rangle,...,\langle v,o_{q+2} \rangle), o_j \in O.$

$$C$$ contains $$q(q-1)^2/2$$ words without 0 entries. The subgraph of the strongly regular graph of $$C$$ induced on the latter words is also strongly regular, assuming $$q>4$$. This is a construction due to A.E.Brouwer [AB16], and leads to graphs with parameters also given by a construction in [HHL09]. According to [AB16], these two constructions are likely to produce isomorphic graphs.

INPUT:

• q – a power of two
• hyperoval – a hyperoval (i.e. a complete 2-arc; a set of points in the plane meeting every line in 0 or 2 points) in $$PG(2,q)$$ over the field field. Each point of hyperoval must be a length 3 vector over field with 1st non-0 coordinate equal to 1. By default, hyperoval and field are not specified, and constructed on the fly. In particular, hyperoval we build is the classical one, i.e. a conic with the point of intersection of its tangent lines.
• field – an instance of a finite field of order $$q$$, must be provided if hyperoval is provided.
• check_hyperoval – (default: True) if True, check hyperoval for correctness.

EXAMPLES:

using the built-in construction:

sage: g=graphs.Nowhere0WordsTwoWeightCodeGraph(8); g
Nowhere0WordsTwoWeightCodeGraph(8): Graph on 196 vertices
sage: g.is_strongly_regular(parameters=True)
(196, 60, 14, 20)
sage: g=graphs.Nowhere0WordsTwoWeightCodeGraph(16) # not tested (long time)
sage: g.is_strongly_regular(parameters=True)       # not tested (long time)
(1800, 728, 268, 312)


sage: F=GF(8)
sage: O=[vector(F,(0,0,1)),vector(F,(0,1,0))]+[vector(F, (1,x^2,x)) for x in F]
sage: g=graphs.Nowhere0WordsTwoWeightCodeGraph(8,hyperoval=O,field=F); g
Nowhere0WordsTwoWeightCodeGraph(8): Graph on 196 vertices
sage: g.is_strongly_regular(parameters=True)
(196, 60, 14, 20)


REFERENCES:

 [HHL09] T. Huang, L. Huang, M.I. Lin On a class of strongly regular designs and quasi-semisymmetric designs. In: Recent Developments in Algebra and Related Areas, ALM vol. 8, pp. 129–153. International Press, Somerville (2009)
 [AB16] (1, 2) A.E. Brouwer Personal communication, 2016
static OctahedralGraph()

Returns an Octahedral graph (with 6 nodes).

The regular octahedron is an 8-sided polyhedron with triangular faces. The octahedral graph corresponds to the connectivity of the vertices of the octahedron. It is the line graph of the tetrahedral graph. The octahedral is symmetric, so the spring-layout algorithm will be very effective for display.

PLOTTING: The Octahedral graph should be viewed in 3 dimensions. We chose to use the default spring-layout algorithm here, so that multiple iterations might yield a different point of reference for the user. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a string argument will be added to select the flat spring-layout over a future implementation.

EXAMPLES: Construct and show an Octahedral graph

sage: g = graphs.OctahedralGraph()
sage: g.show() # long time


Create several octahedral graphs in a Sage graphics array They will be drawn differently due to the use of the spring-layout algorithm

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.OctahedralGraph()
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static OddGraph(n)

Returns the Odd Graph with parameter $$n$$.

The Odd Graph with parameter $$n$$ is defined as the Kneser Graph with parameters $$2n-1,n-1$$. Equivalently, the Odd Graph is the graph whose vertices are the $$n-1$$-subsets of $$[0,1,\dots,2(n-1)]$$, and such that two vertices are adjacent if their corresponding sets are disjoint.

For example, the Petersen Graph can be defined as the Odd Graph with parameter $$3$$.

EXAMPLES:

sage: OG = graphs.OddGraph(3)
sage: sorted(OG.vertex_iterator(), key=str)
[{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5},
{3, 4}, {3, 5}, {4, 5}]
sage: P = graphs.PetersenGraph()
sage: P.is_isomorphic(OG)
True

static OrthogonalArrayBlockGraph(k, n, OA=None)

Returns the graph of an $$OA(k,n)$$.

The intersection graph of the blocks of a transversal design with parameters $$(k,n)$$, or $$TD(k,n)$$ for short, is a strongly regular graph (unless it is a complete graph). Its parameters $$(v,k',\lambda,\mu)$$ are determined by the parameters $$k,n$$ via:

$v=n^2, k'=k(n-1), \lambda=(k-1)(k-2)+n-2, \mu=k(k-1)$

As transversal designs and orthogonal arrays (OA for short) are equivalent objects, this graph can also be built from the blocks of an $$OA(k,n)$$, two of them being adjacent if one of their coordinates match.

For more information on these graphs, see Andries Brouwer’s page on Orthogonal Array graphs.

Warning

• Brouwer’s website uses the notation $$OA(n,k)$$ instead of $$OA(k,n)$$
• For given parameters $$k$$ and $$n$$ there can be many $$OA(k,n)$$ : the graphs returned are not uniquely defined by their parameters (see the examples below).
• If the function is called only with the parameter k and n the results might be different with two versions of Sage, or even worse : some could not be available anymore.

INPUT:

• k,n (integers)
• OA – An orthogonal array. If set to None (default) then orthogonal_array() is called to compute an $$OA(k,n)$$.

EXAMPLES:

sage: G = graphs.OrthogonalArrayBlockGraph(5,5); G
OA(5,5): Graph on 25 vertices
sage: G.is_strongly_regular(parameters=True)
(25, 20, 15, 20)
sage: G = graphs.OrthogonalArrayBlockGraph(4,10); G
OA(4,10): Graph on 100 vertices
sage: G.is_strongly_regular(parameters=True)
(100, 36, 14, 12)


Two graphs built from different orthogonal arrays are also different:

sage: k=4;n=10
sage: OAa = designs.orthogonal_arrays.build(k,n)
sage: OAb = [[(x+1)%n for x in R] for R in OAa]
sage: set(map(tuple,OAa)) == set(map(tuple,OAb))
False
sage: Ga = graphs.OrthogonalArrayBlockGraph(k,n,OAa)
sage: Gb = graphs.OrthogonalArrayBlockGraph(k,n,OAb)
sage: Ga == Gb
False


As OAb was obtained from OAa by a relabelling the two graphs are isomorphic:

sage: Ga.is_isomorphic(Gb)
True


But there are examples of $$OA(k,n)$$ for which the resulting graphs are not isomorphic:

sage: oa0 = [[0, 0, 1], [0, 1, 3], [0, 2, 0], [0, 3, 2],
....:        [1, 0, 3], [1, 1, 1], [1, 2, 2], [1, 3, 0],
....:        [2, 0, 0], [2, 1, 2], [2, 2, 1], [2, 3, 3],
....:        [3, 0, 2], [3, 1, 0], [3, 2, 3], [3, 3, 1]]
sage: oa1 = [[0, 0, 1], [0, 1, 0], [0, 2, 3], [0, 3, 2],
....:        [1, 0, 3], [1, 1, 2], [1, 2, 0], [1, 3, 1],
....:        [2, 0, 0], [2, 1, 1], [2, 2, 2], [2, 3, 3],
....:        [3, 0, 2], [3, 1, 3], [3, 2, 1], [3, 3, 0]]
sage: g0 = graphs.OrthogonalArrayBlockGraph(3,4,oa0)
sage: g1 = graphs.OrthogonalArrayBlockGraph(3,4,oa1)
sage: g0.is_isomorphic(g1)
False


But nevertheless isospectral:

sage: g0.spectrum()
[9, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, -3, -3, -3, -3, -3]
sage: g1.spectrum()
[9, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3, -3, -3, -3, -3, -3]


Note that the graph g0 is actually isomorphic to the affine polar graph $$VO^+(4,2)$$:

sage: graphs.AffineOrthogonalPolarGraph(4,2,'+').is_isomorphic(g0)
True

static OrthogonalPolarGraph(m, q, sign='+')

Returns the Orthogonal Polar Graph $$O^{\epsilon}(m,q)$$.

For more information on Orthogonal Polar graphs, see the page of Andries Brouwer’s website.

INPUT:

• m,q (integers) – $$q$$ must be a prime power.
• sign"+" or "-" if $$m$$ is even, "+" (default) otherwise.

EXAMPLES:

sage: G = graphs.OrthogonalPolarGraph(6,3,"+"); G
Orthogonal Polar Graph O^+(6, 3): Graph on 130 vertices
sage: G.is_strongly_regular(parameters=True)
(130, 48, 20, 16)
sage: G = graphs.OrthogonalPolarGraph(6,3,"-"); G
Orthogonal Polar Graph O^-(6, 3): Graph on 112 vertices
sage: G.is_strongly_regular(parameters=True)
(112, 30, 2, 10)
sage: G = graphs.OrthogonalPolarGraph(5,3); G
Orthogonal Polar Graph O(5, 3): Graph on 40 vertices
sage: G.is_strongly_regular(parameters=True)
(40, 12, 2, 4)
sage: G = graphs.OrthogonalPolarGraph(8,2,"+"); G
Orthogonal Polar Graph O^+(8, 2): Graph on 135 vertices
sage: G.is_strongly_regular(parameters=True)
(135, 70, 37, 35)
sage: G = graphs.OrthogonalPolarGraph(8,2,"-"); G
Orthogonal Polar Graph O^-(8, 2): Graph on 119 vertices
sage: G.is_strongly_regular(parameters=True)
(119, 54, 21, 27)

static PaleyGraph(q)

Paley graph with $$q$$ vertices

Parameter $$q$$ must be the power of a prime number and congruent to 1 mod 4.

EXAMPLES:

sage: G = graphs.PaleyGraph(9); G
Paley graph with parameter 9: Graph on 9 vertices
sage: G.is_regular()
True


A Paley graph is always self-complementary:

sage: G.is_self_complementary()
True

static PappusGraph()

Return the Pappus graph, a graph on 18 vertices.

The Pappus graph is cubic, symmetric, and distance-regular.

EXAMPLES:

sage: G = graphs.PappusGraph()
sage: G.show()  # long time
sage: L = graphs.LCFGraph(18, [5,7,-7,7,-7,-5], 3)
sage: L.show()  # long time
sage: G.is_isomorphic(L)
True

static PasechnikGraph(n)

Pasechnik strongly regular graph on $$(4n-1)^2$$ vertices

A strongly regular graph with parameters of the orthogonal array graph OrthogonalArrayBlockGraph(), also known as pseudo Latin squares graph $$L_{2n-1}(4n-1)$$, constructed from a skew Hadamard matrix of order $$4n$$ following [Pa92].

EXAMPLES:

sage: graphs.PasechnikGraph(4).is_strongly_regular(parameters=True)
(225, 98, 43, 42)
sage: graphs.PasechnikGraph(9).is_strongly_regular(parameters=True) # long time
(1225, 578, 273, 272)

static PathGraph(n, pos=None)

Return a path graph with $$n$$ nodes.

A path graph is a graph where all inner nodes are connected to their two neighbors and the two end-nodes are connected to their one inner neighbors (i.e.: a cycle graph without the first and last node connected).

INPUT:

• n – number of nodes of the path graph
• pos (default: None) – a string which is either ‘circle’ or ‘line’ (otherwise the default is used) indicating which embedding algorithm to use. See the plotting section below for more detail.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the graph may be drawn in one of two ways: The ‘line’ argument will draw the graph in a horizontal line (left to right) if there are less than 11 nodes. Otherwise the ‘line’ argument will append horizontal lines of length 10 nodes below, alternating left to right and right to left. The ‘circle’ argument will cause the graph to be drawn in a cycle-shape, with the first node at the top and then about the circle in a clockwise manner. By default (without an appropriate string argument) the graph will be drawn as a ‘circle’ if $$10 < n < 41$$ and as a ‘line’ for all other $$n$$.

EXAMPLES: Show default drawing by size: ‘line’: $$n \leq 10$$

sage: p = graphs.PathGraph(10)
sage: p.show() # long time


‘circle’: $$10 < n < 41$$

sage: q = graphs.PathGraph(25)
sage: q.show() # long time


‘line’: $$n \geq 41$$

sage: r = graphs.PathGraph(55)
sage: r.show() # long time


Override the default drawing:

sage: s = graphs.PathGraph(5,'circle')
sage: s.show() # long time

static PerkelGraph()

Return the Perkel Graph.

The Perkel Graph is a 6-regular graph with $$57$$ vertices and $$171$$ edges. It is the unique distance-regular graph with intersection array $$(6,5,2;1,1,3)$$. For more information, see the Wikipedia article Perkel_graph or https://www.win.tue.nl/~aeb/graphs/Perkel.html.

EXAMPLES:

sage: g = graphs.PerkelGraph(); g
Perkel Graph: Graph on 57 vertices
sage: g.is_distance_regular(parameters=True)
([6, 5, 2, None], [None, 1, 1, 3])

static PermutationGraph(second_permutation, first_permutation=None)

Build a permutation graph from one permutation or from two lists.

Definition:

If $$\sigma$$ is a permutation of $$\{ 1, 2, \ldots, n \}$$, then the permutation graph of $$\sigma$$ is the graph on vertex set $$\{ 1, 2, \ldots, n \}$$ in which two vertices $$i$$ and $$j$$ satisfying $$i < j$$ are connected by an edge if and only if $$\sigma^{-1}(i) > \sigma^{-1}(j)$$. A visual way to construct this graph is as follows:

Take two horizontal lines in the euclidean plane, and mark points $$1, ..., n$$ from left to right on the first of them. On the second one, still from left to right, mark $$n$$ points $$\sigma(1), \sigma(2), \ldots, \sigma(n)$$. Now, link by a segment the two points marked with $$1$$, then link together the points marked with $$2$$, and so on. The permutation graph of $$\sigma$$ is the intersection graph of those segments: there exists a vertex in this graph for each element from $$1$$ to $$n$$, two vertices $$i, j$$ being adjacent if the segments $$i$$ and $$j$$ cross each other.

The set of edges of the permutation graph can thus be identified with the set of inversions of the inverse of the given permutation $$\sigma$$.

A more general notion of permutation graph can be defined as follows: If $$S$$ is a set, and $$(a_1, a_2, \ldots, a_n)$$ and $$(b_1, b_2, \ldots, b_n)$$ are two lists of elements of $$S$$, each of which lists contains every element of $$S$$ exactly once, then the permutation graph defined by these two lists is the graph on the vertex set $$S$$ in which two vertices $$i$$ and $$j$$ are connected by an edge if and only if the order in which these vertices appear in the list $$(a_1, a_2, \ldots, a_n)$$ is the opposite of the order in which they appear in the list $$(b_1, b_2, \ldots, b_n)$$. When $$(a_1, a_2, \ldots, a_n) = (1, 2, \ldots, n)$$, this graph is the permutation graph of the permutation $$(b_1, b_2, \ldots, b_n) \in S_n$$. Notice that $$S$$ does not have to be a set of integers here, but can be a set of strings, tuples, or anything else. We can still use the above visual description to construct the permutation graph, but now we have to mark points $$a_1, a_2, \ldots, a_n$$ from left to right on the first horizontal line and points $$b_1, b_2, \ldots, b_n$$ from left to right on the second horizontal line.

INPUT:

• second_permutation – the unique permutation/list defining the graph, or the second of the two (if the graph is to be built from two permutations/lists).

• first_permutation (optional) – the first of the two permutations/lists from which the graph should be built, if it is to be built from two permutations/lists.

When first_permutation is None (default), it is set to be equal to sorted(second_permutation), which yields the expected ordering when the elements of the graph are integers.

EXAMPLES:

sage: p = Permutations(5).random_element()
sage: PG = graphs.PermutationGraph(p)
sage: edges = PG.edges(labels=False)
sage: set(edges) == set(p.inverse().inversions())
True

sage: PG = graphs.PermutationGraph([3,4,5,1,2])
sage: sorted(PG.edges())
[(1, 3, None),
(1, 4, None),
(1, 5, None),
(2, 3, None),
(2, 4, None),
(2, 5, None)]
sage: PG = graphs.PermutationGraph([3,4,5,1,2], [1,4,2,5,3])
sage: sorted(PG.edges())
[(1, 3, None),
(1, 4, None),
(1, 5, None),
(2, 3, None),
(2, 5, None),
(3, 4, None),
(3, 5, None)]
sage: PG = graphs.PermutationGraph([1,4,2,5,3], [3,4,5,1,2])
sage: sorted(PG.edges())
[(1, 3, None),
(1, 4, None),
(1, 5, None),
(2, 3, None),
(2, 5, None),
(3, 4, None),
(3, 5, None)]

sage: PG = graphs.PermutationGraph(Permutation([1,3,2]), Permutation([1,2,3]))
sage: sorted(PG.edges())
[(2, 3, None)]

sage: graphs.PermutationGraph([]).edges()
[]
sage: graphs.PermutationGraph([], []).edges()
[]

sage: PG = graphs.PermutationGraph("graph", "phrag")
sage: sorted(PG.edges())
[('a', 'g', None),
('a', 'h', None),
('a', 'p', None),
('g', 'h', None),
('g', 'p', None),
('g', 'r', None),
('h', 'r', None),
('p', 'r', None)]

static PetersenGraph()

Return the Petersen Graph.

The Petersen Graph is a named graph that consists of 10 vertices and 15 edges, usually drawn as a five-point star embedded in a pentagon.

The Petersen Graph is a common counterexample. For example, it is not Hamiltonian.

PLOTTING: See the plotting section for the generalized Petersen graphs.

EXAMPLES: We compare below the Petersen graph with the default spring-layout versus a planned position dictionary of [x,y] tuples:

sage: petersen_spring = Graph({0:[1,4,5], 1:[0,2,6], 2:[1,3,7], 3:[2,4,8], 4:[0,3,9], 5:[0,7,8], 6:[1,8,9], 7:[2,5,9], 8:[3,5,6], 9:[4,6,7]})
sage: petersen_spring.show() # long time
sage: petersen_database = graphs.PetersenGraph()
sage: petersen_database.show() # long time

static PoussinGraph()

Return the Poussin Graph.

For more information on the Poussin Graph, see its corresponding Wolfram page.

EXAMPLES:

sage: g = graphs.PoussinGraph()
sage: g.order()
15
sage: g.is_planar()
True

static QueenGraph(dim_list, radius=None, relabel=False)

Returns the $$d$$-dimensional Queen Graph with prescribed dimensions.

The 2-dimensional Queen Graph of parameters $$n$$ and $$m$$ is a graph with $$nm$$ vertices in which each vertex represents a square in an $$n \times m$$ chessboard, and each edge corresponds to a legal move by a queen.

The $$d$$-dimensional Queen Graph with $$d >= 2$$ has for vertex set the cells of a $$d$$-dimensional grid with prescribed dimensions, and each edge corresponds to a legal move by a queen in either one or two dimensions.

All 2-dimensional Queen Graphs are Hamiltonian and biconnected. The chromatic number of a $$(n,n)$$-Queen Graph is at least $$n$$, and it is exactly $$n$$ when $$n\equiv 1,5 \bmod{6}$$.

INPUT:

• dim_list – an iterable object (list, set, dict) providing the dimensions $$n_1, n_2, \ldots, n_d$$, with $$n_i \geq 1$$, of the chessboard.
• radius – (default: None) by setting the radius to a positive integer, one may reduce the visibility of the queen to at most radius steps. When radius is 1, the resulting graph is a King Graph.
• relabel – (default: False) a boolean set to True if vertices must be relabeled as integers.

EXAMPLES:

The $$(2,2)$$-Queen Graph is isomorphic to the complete graph on 4 vertices:

sage: G = graphs.QueenGraph( [2, 2] )
sage: G.is_isomorphic( graphs.CompleteGraph(4) )
True


The Queen Graph with radius 1 is isomorphic to the King Graph:

sage: G = graphs.QueenGraph( [4, 5], radius=1 )
sage: H = graphs.KingGraph( [5, 4] )
sage: G.is_isomorphic( H )
True


Also True in higher dimensions:

sage: G = graphs.QueenGraph( [3, 4, 5], radius=1 )
sage: H = graphs.KingGraph( [5, 3, 4] )
sage: G.is_isomorphic( H )
True


The Queen Graph can be obtained from the Rook Graph and the Bishop Graph:

sage: for d in range(3,12):   # long time
....:     for r in range(1,d+1):
....:         if not G.is_isomorphic(H):
....:            print("that's not good!")

static RandomBarabasiAlbert(n, m, seed=None)

Return a random graph created using the Barabasi-Albert preferential attachment model.

A graph with $$m$$ vertices and no edges is initialized, and a graph of $$n$$ vertices is grown by attaching new vertices each with $$m$$ edges that are attached to existing vertices, preferentially with high degree.

INPUT:

• n – number of vertices in the graph
• m – number of edges to attach from each new node
• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

We show the edge list of a random graph on 6 nodes with $$m = 2$$:

sage: G = graphs.RandomBarabasiAlbert(6,2)
sage: G.order(), G.size()
(6, 8)
sage: G.degree_sequence()  # random
[4, 3, 3, 2, 2, 2]


We plot a random graph on 12 nodes with $$m = 3$$:

sage: ba = graphs.RandomBarabasiAlbert(12,3)
sage: ba.show()  # long time


We view many random graphs using a graphics array:

sage: g = []
sage: j = []
sage: for i in range(1,10):
....:     k = graphs.RandomBarabasiAlbert(i+3, 3)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show()  # long time


When $$m = 1$$, the generated graph is a tree:

sage: graphs.RandomBarabasiAlbert(6, 1).is_tree()
True

static RandomBicubicPlanar(n)

Return the graph of a random bipartite cubic map with $$3 n$$ edges.

INPUT:

$$n$$ – an integer (at least $$1$$)

OUTPUT:

a graph with multiple edges (no embedding is provided)

The algorithm used is described in [Schaeffer99]. This samples a random rooted bipartite cubic map, chosen uniformly at random.

First one creates a random binary tree with $$n$$ vertices. Next one turns this into a blossoming tree (at random) and reads the contour word of this blossoming tree.

Then one performs a rotation on this word so that this becomes a balanced word. There are three ways to do that, one is picked at random. Then a graph is build from the balanced word by iterated closure (adding edges).

In the returned graph, the three edges incident to any given vertex are colored by the integers 0, 1 and 2.

the auxiliary method blossoming_contour()

EXAMPLES:

sage: n = randint(200, 300)
sage: G = graphs.RandomBicubicPlanar(n)
sage: G.order() == 2*n
True
sage: G.size() == 3*n
True
sage: G.is_bipartite() and G.is_planar() and G.is_regular(3)
True
sage: dic = {'red':[v for v in G.vertices() if v == 'n'],
....:        'blue': [v for v in G.vertices() if v != 'n']}
sage: G.plot(vertex_labels=False,vertex_size=20,vertex_colors=dic)
Graphics object consisting of ... graphics primitives


REFERENCES:

 [Schaeffer99] Gilles Schaeffer, Random Sampling of Large Planar Maps and Convex Polyhedra, Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999)
static RandomBipartite(n1, n2, p, set_position=False)

Returns a bipartite graph with $$n1+n2$$ vertices such that any edge from $$[n1]$$ to $$[n2]$$ exists with probability $$p$$.

INPUT:

• n1, n2 – Cardinalities of the two sets
• p – Probability for an edge to exist
• set_position – boolean (default False); if set to True, we assign positions to the vertices so that the set of cardinality $$n1$$ is on the line $$y=1$$ and the set of cardinality $$n2$$ is on the line $$y=0$$.

EXAMPLES:

sage: g = graphs.RandomBipartite(5, 2, 0.5)
sage: g.vertices()
[(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 1)]

static RandomBlockGraph(m, k, kmax=None, incidence_structure=False)

Return a Random Block Graph.

A block graph is a connected graph in which every biconnected component (block) is a clique.

INPUT:

• m – integer; number of blocks (at least one).
• k – integer; minimum number of vertices of a block (at least two).
• kmax – integer (default: None) By default, each block has $$k$$ vertices. When the parameter $$kmax$$ is specified (with $$kmax \geq k$$), the number of vertices of each block is randomly chosen between $$k$$ and $$kmax$$.
• incidence_structure – boolean (default: False) when set to True, the incidence structure of the graphs is returned instead of the graph itself, that is the list of the lists of vertices in each block. This is useful for the creation of some hypergraphs.

OUTPUT:

A Graph when incidence_structure==False (default), and otherwise an incidence structure.

EXAMPLES:

A block graph with a single block is a clique:

sage: B = graphs.RandomBlockGraph(1, 4)
sage: B.is_clique()
True


A block graph with blocks of order 2 is a tree:

sage: B = graphs.RandomBlockGraph(10, 2)
sage: B.is_tree()
True


Every biconnected component of a block graph is a clique:

sage: B = graphs.RandomBlockGraph(5, 3, kmax=6)
sage: blocks,cuts = B.blocks_and_cut_vertices()
sage: all(B.is_clique(block) for block in blocks)
True


A block graph with blocks of order $$k$$ has $$m*(k-1)+1$$ vertices:

sage: m, k = 6, 4
sage: B = graphs.RandomBlockGraph(m, k)
sage: B.order() == m*(k-1)+1
True


Test recognition methods:

sage: B = graphs.RandomBlockGraph(6, 2, kmax=6)
sage: B.is_block_graph()
True
sage: B in graph_classes.Block
True


sage: m, k = 6, 4
sage: IS = graphs.RandomBlockGraph(m, k, incidence_structure=True)
sage: from sage.combinat.designs.incidence_structures import IncidenceStructure
sage: IncidenceStructure(IS)
Incidence structure with 19 points and 6 blocks
sage: m*(k-1)+1
19

static RandomBoundedToleranceGraph(n)

Returns a random bounded tolerance graph.

The random tolerance graph is built from a random bounded tolerance representation by using the function $$ToleranceGraph$$. This representation is a list $$((l_0,r_0,t_0), (l_1,r_1,t_1), ..., (l_k,r_k,t_k))$$ where $$k = n-1$$ and $$I_i = (l_i,r_i)$$ denotes a random interval and $$t_i$$ a random positive value less then or equal to the length of the interval $$I_i$$. The width of the representation is limited to n**2 * 2**n.

Note

The tolerance representation used to create the graph can be recovered using get_vertex() or get_vertices().

INPUT:

• n – number of vertices of the random graph.

EXAMPLES:

Every (bounded) tolerance graph is perfect. Hence, the chromatic number is equal to the clique number

sage: g = graphs.RandomBoundedToleranceGraph(8)
sage: g.clique_number() == g.chromatic_number()
True

static RandomChordalGraph(n, algorithm='growing', k=None, l=None, f=None, s=None)

Return a random chordal graph of order n.

A Graph $$G$$ is said to be chordal if it contains no induced hole (a cycle of length at least 4). Equivalently, $$G$$ is chordal if it has a perfect elimination orderings, if each minimal separator is a clique, or if it is the intersection graphs of subtrees of a tree. See the Wikipedia article Chordal_graph.

This generator implements the algorithms proposed in [SHET2018] for generating random chordal graphs as the intersection graph of $$n$$ subtrees of a tree of order $$n$$.

The returned graph is not necessarily connected.

INPUT:

• n – integer; the number of nodes of the graph
• algorithm – string (default: "growing"); the choice of the algorithm for randomly selecting $$n$$ subtrees of a random tree of order $$n$$. Possible choices are:
• "growing" – for each subtree $$T_i$$, the algorithm picks a size $$k_i$$ randomly from $$[1,k]$$. Then a random node of $$T$$ is chosen as the first node of $$T_i$$. In each of the subsequent $$k_i - 1$$ iterations, it picks a random node in the neighborhood of $$T_i$$ and adds it to $$T_i$$.
• "connecting" – for each subtree $$T_i$$, it first selects $$k_i$$ nodes of $$T$$, where $$k_i$$ is a random integer from a Poisson distribution with mean $$l$$. $$T_i$$ is then generated to be the minimal subtree containing the selected $$k_i$$ nodes. This implies that a subtree will most likely have many more nodes than those selected initially, and this must be taken into consideration when choosing $$l$$.
• "pruned" – for each subtree $$T_i$$, it randomly selects a fraction $$f$$ of the edges on the tree and removes them. The number of edges to delete, say $$l$$, is calculated as $$\lfloor (n - 1) f \rfloor$$, which will leave $$l + 1$$ subtrees in total. Then, it determines the sizes of the $$l + 1$$ subtrees and stores the distinct values. Finally, it picks a random size $$k_i$$ from the set of largest $$100(1-s)\%$$ of distinct values, and randomly chooses a subtree with size $$k_i$$.
• k – integer (default: None); maximum size of a subtree. If not specified (None), the maximum size is set to $$\sqrt{n}$$. This parameter is used only when algorithm="growing". See growing_subtrees() for more details.
• l – a strictly positive real number (default: None); mean of a Poisson distribution. If not specified, the mean in set to $$\log_2{n}$$. This parameter is used only when algorithm="connecting". See connecting_nodes() for more details.
• f – a rational number (default: None); the edge deletion fraction. This value must be choosen in $$[0..1]$$. If not specified, this parameter is set to $$\frac{1}{n-1}$$. This parameter is used only when algorithm="pruned". See pruned_tree() for more details.
• s – a real number between 0 and 1 (default: None); selection barrier for the size of trees. If not specified, this parameter is set to $$0.5$$. This parameter is used only when algorithm="pruned". See pruned_tree() for more details.

EXAMPLES:

sage: from sage.graphs.generators.random import RandomChordalGraph
sage: T = RandomChordalGraph(20, algorithm="growing", k=5)
sage: T.is_chordal()
True
sage: T = RandomChordalGraph(20, algorithm="connecting", l=3)
sage: T.is_chordal()
True
sage: T = RandomChordalGraph(20, algorithm="pruned", f=1/3, s=.5)
sage: T.is_chordal()
True


static RandomGNM(n, m, dense=False, seed=None)

Returns a graph randomly picked out of all graphs on n vertices with m edges.

INPUT:

• n - number of vertices.
• m - number of edges.
• dense - whether to use NetworkX’s dense_gnm_random_graph or gnm_random_graph
• seed - a random.Random seed or a Python int for the random number generator (default: None).

EXAMPLES: We show the edge list of a random graph on 5 nodes with 10 edges.

sage: graphs.RandomGNM(5, 10).edges(labels=False)
[(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)]


We plot a random graph on 12 nodes with m = 12.

sage: gnm = graphs.RandomGNM(12, 12)
sage: gnm.show()  # long time


We view many random graphs using a graphics array:

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.RandomGNM(i+3, i^2-i)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show()  # long time

static RandomGNP(n, p, seed=None, fast=True, algorithm='Sage')

Returns a random graph on $$n$$ nodes. Each edge is inserted independently with probability $$p$$.

INPUT:

• n – number of nodes of the graph
• p – probability of an edge
• seed - a random.Random seed or a Python int for the random number generator (default: None).
• fast – boolean set to True (default) to use the algorithm with time complexity in $$O(n+m)$$ proposed in [BatBra2005]. It is designed for generating large sparse graphs. It is faster than other algorithms for LARGE instances (try it to know whether it is useful for you).
• algorithm – By default (algorithm='Sage'), this function uses the algorithm implemented in sage.graphs.graph_generators_pyx.pyx. When algorithm='networkx', this function calls the NetworkX function fast_gnp_random_graph, unless fast=False, then gnp_random_graph. Try them to know which algorithm is the best for you. The fast parameter is not taken into account by the ‘Sage’ algorithm so far.

REFERENCES:

 [ErdRen1959] P. Erdos and A. Renyi. On Random Graphs, Publ. Math. 6, 290 (1959).
 [Gilbert1959] E. N. Gilbert. Random Graphs, Ann. Math. Stat., 30, 1141 (1959).
 [BatBra2005] V. Batagelj and U. Brandes. Efficient generation of large random networks. Phys. Rev. E, 71, 036113, 2005.

PLOTTING: When plotting, this graph will use the default spring-layout algorithm, unless a position dictionary is specified.

EXAMPLES: We show the edge list of a random graph on 6 nodes with probability $$p = .4$$:

sage: set_random_seed(0)
sage: graphs.RandomGNP(6, .4).edges(labels=False)
[(0, 1), (0, 5), (1, 2), (2, 4), (3, 4), (3, 5), (4, 5)]


We plot a random graph on 12 nodes with probability $$p = .71$$:

sage: gnp = graphs.RandomGNP(12,.71)
sage: gnp.show() # long time


We view many random graphs using a graphics array:

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.RandomGNP(i+3,.43)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time
sage: graphs.RandomGNP(4,1)
Complete graph: Graph on 4 vertices

static RandomHolmeKim(n, m, p, seed=None)

Return a random graph generated by the Holme and Kim algorithm for graphs with power law degree distribution and approximate average clustering.

INPUT:

• n – number of vertices
• m – number of random edges to add for each new node
• p – probability of adding a triangle after adding a random edge
• seed – a random.Random seed or a Python int for the random number generator (default: None)

From the NetworkX documentation: the average clustering has a hard time getting above a certain cutoff that depends on $$m$$. This cutoff is often quite low. Note that the transitivity (fraction of triangles to possible triangles) seems to go down with network size. It is essentially the Barabasi-Albert growth model with an extra step that each random edge is followed by a chance of making an edge to one of its neighbors too (and thus a triangle). This algorithm improves on B-A in the sense that it enables a higher average clustering to be attained if desired. It seems possible to have a disconnected graph with this algorithm since the initial $$m$$ nodes may not be all linked to a new node on the first iteration like the BA model.

EXAMPLES:

We check that a random graph on 8 nodes with 2 random edges per node and a probability $$p = 0.5$$ of forming triangles contains a triangle:

sage: G = graphs.RandomHolmeKim(8, 2, 0.5)
sage: G.order(), G.size()
(8, 12)
sage: C3 = graphs.CycleGraph(3)
sage: G.subgraph_search(C3)
Subgraph of (): Graph on 3 vertices

sage: G = graphs.RandomHolmeKim(12, 3, .3)
sage: G.show()  # long time


REFERENCE:

 [HolmeKim2002] Holme, P. and Kim, B.J. Growing scale-free networks with tunable clustering, Phys. Rev. E (2002). vol 65, no 2, 026107.
static RandomIntervalGraph(n)

Returns a random interval graph.

An interval graph is built from a list $$(a_i,b_i)_{1\leq i \leq n}$$ of intervals : to each interval of the list is associated one vertex, two vertices being adjacent if the two corresponding intervals intersect.

A random interval graph of order $$n$$ is generated by picking random values for the $$(a_i,b_j)$$, each of the two coordinates being generated from the uniform distribution on the interval $$[0,1]$$.

This definitions follows [boucheron2001].

Note

The vertices are named 0, 1, 2, and so on. The intervals used to create the graph are saved with the graph and can be recovered using get_vertex() or get_vertices().

INPUT:

• n (integer) – the number of vertices in the random graph.

EXAMPLES:

As for any interval graph, the chromatic number is equal to the clique number

sage: g = graphs.RandomIntervalGraph(8)
sage: g.clique_number() == g.chromatic_number()
True


REFERENCE:

 [boucheron2001] Boucheron, S. and FERNANDEZ de la VEGA, W., On the Independence Number of Random Interval Graphs, Combinatorics, Probability and Computing v10, issue 05, Pages 385–396, Cambridge Univ Press, 2001
static RandomLobster(n, p, q, seed=None)

Returns a random lobster.

A lobster is a tree that reduces to a caterpillar when pruning all leaf vertices. A caterpillar is a tree that reduces to a path when pruning all leaf vertices (q=0).

INPUT:

• n - expected number of vertices in the backbone
• p - probability of adding an edge to the backbone
• q - probability of adding an edge (claw) to the arms
• seed - a random.Random seed or a Python int for the random number generator (default: None).

EXAMPLES: We show the edge list of a random graph with 3 backbone nodes and probabilities $$p = 0.7$$ and $$q = 0.3$$:

sage: graphs.RandomLobster(3, 0.7, 0.3).edges(labels=False)
[]                                                                  # 32-bit
[(0, 1), (0, 5), (1, 2), (1, 6), (2, 3), (2, 7), (3, 4), (3, 8)]    # 64-bit

sage: G = graphs.RandomLobster(9, .6, .3)
sage: G.show()  # long time

static RandomNewmanWattsStrogatz(n, k, p, seed=None)

Return a Newman-Watts-Strogatz small world random graph on $$n$$ vertices.

From the NetworkX documentation: first create a ring over $$n$$ nodes. Then each node in the ring is connected with its $$k$$ nearest neighbors. Then shortcuts are created by adding new edges as follows: for each edge $$u-v$$ in the underlying “$$n$$-ring with $$k$$ nearest neighbors”; with probability $$p$$ add a new edge $$u-w$$ with randomly-chosen existing node $$w$$. In contrast with networkx.watts_strogatz_graph(), no edges are removed.

INPUT:

• n – number of vertices
• k – each vertex is connected to its $$k$$ nearest neighbors
• p – the probability of adding a new edge for each edge
• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

We check that the generated graph contains a cycle of order $$n$$:

sage: G = graphs.RandomNewmanWattsStrogatz(7, 2, 0.2)
sage: G.order(), G.size()
(7, 9)
sage: C7 = graphs.CycleGraph(7)
sage: G.subgraph_search(C7)
Subgraph of (): Graph on 7 vertices
sage: G.diameter() <= C7.diameter()
True

sage: G = graphs.RandomNewmanWattsStrogatz(12, 2, .3)
sage: G.show()  # long time


REFERENCE:

 [NWS99] Newman, M.E.J., Watts, D.J. and Strogatz, S.H. Random graph models of social networks. Proc. Nat. Acad. Sci. USA 99, 2566-2572.
static RandomRegular(d, n, seed=None)

Return a random $$d$$-regular graph on $$n$$ vertices, or False on failure.

Since every edge is incident to two vertices, $$n\times d$$ must be even.

INPUT:

• d – degree
• n – number of vertices
• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

We check that a random graph with 8 nodes each of degree 3 is 3-regular:

sage: G = graphs.RandomRegular(3, 8)
sage: G.is_regular(k=3)
True
sage: G.degree_histogram()
[0, 0, 0, 8]

sage: G = graphs.RandomRegular(3, 20)
sage: if G:
....:     G.show()  # random output, long time


REFERENCES:

 [KimVu2003] Kim, Jeong Han and Vu, Van H. Generating random regular graphs. Proc. 35th ACM Symp. on Thy. of Comp. 2003, pp 213-222. ACM Press, San Diego, CA, USA. http://doi.acm.org/10.1145/780542.780576
 [StegerWormald1999] Steger, A. and Wormald, N. Generating random regular graphs quickly. Prob. and Comp. 8 (1999), pp 377-396.
static RandomRegularBipartite(n1, n2, d1, set_position=False)

Return a random regular bipartite graph on $$n1 + n2$$ vertices.

The bipartite graph has $$n1 * d1$$ edges. Hence, $$n2$$ must divide $$n1 * d1$$. Each vertex of the set of cardinality $$n1$$ has degree $$d1$$ (which can be at most $$n2$$) and each vertex in the set of cardinality $$n2$$ has degree $$(n1 * d1) / n2$$. The bipartite graph has no multiple edges.

This generator implements an algorithm inspired by that of [MW1990] for the uniform generation of random regular bipartite graphs. It performs well when $$d1 = o(n2^{1/3})$$ or ($$n2 - d1 = o(n2^{1/3})$$). In other cases, the running time can be huge. Note that the currently implemented algorithm does not generate uniformly random graphs.

INPUT:

• n1, n2 – number of vertices in each side
• d1 – degree of the vertices in the set of cardinality $$n1$$.
• set_position – boolean (default False); if set to True, we assign positions to the vertices so that the set of cardinality $$n1$$ is on the line $$y=1$$ and the set of cardinality $$n2$$ is on the line $$y=0$$.

EXAMPLES:

sage: g = graphs.RandomRegularBipartite(4, 6, 3)
sage: g.order(), g.size()
(10, 12)
sage: set(g.degree())
{2, 3}

sage: graphs.RandomRegularBipartite(1, 2, 2, set_position=True).get_pos()
{0: (1, 1.0), 1: (0, 0), 2: (2.0, 0.0)}
sage: graphs.RandomRegularBipartite(2, 1, 1, set_position=True).get_pos()
{0: (0, 1), 1: (2.0, 1.0), 2: (1, 0.0)}
sage: graphs.RandomRegularBipartite(2, 3, 3, set_position=True).get_pos()
{0: (0, 1), 1: (3.0, 1.0), 2: (0, 0), 3: (1.5, 0.0), 4: (3.0, 0.0)}
sage: graphs.RandomRegularBipartite(2, 3, 3, set_position=False).get_pos()

static RandomShell(constructor, seed=None)

Return a random shell graph for the constructor given.

INPUT:

• constructor – a list of 3-tuples $$(n, m, d)$$, each representing a shell, where:
• n – the number of vertices in the shell
• m – the number of edges in the shell
• d – the ratio of inter (next) shell edges to intra shell edges
• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

sage: G = graphs.RandomShell([(10,20,0.8),(20,40,0.8)])
sage: G.order(), G.size()
(30, 52)
sage: G.show()  # long time

static RandomToleranceGraph(n)

Returns a random tolerance graph.

The random tolerance graph is built from a random tolerance representation by using the function $$ToleranceGraph$$. This representation is a list $$((l_0,r_0,t_0), (l_1,r_1,t_1), ..., (l_k,r_k,t_k))$$ where $$k = n-1$$ and $$I_i = (l_i,r_i)$$ denotes a random interval and $$t_i$$ a random positive value. The width of the representation is limited to n**2 * 2**n.

Note

The vertices are named 0, 1, …, n-1. The tolerance representation used to create the graph is saved with the graph and can be recovered using get_vertex() or get_vertices().

INPUT:

• n – number of vertices of the random graph.

EXAMPLES:

Every tolerance graph is perfect. Hence, the chromatic number is equal to the clique number

sage: g = graphs.RandomToleranceGraph(8)
sage: g.clique_number() == g.chromatic_number()
True

static RandomTree(n)

Returns a random tree on $$n$$ nodes numbered $$0$$ through $$n-1$$.

By Cayley’s theorem, there are $$n^{n-2}$$ trees with vertex set $$\{0,1,...,n-1\}$$. This constructor chooses one of these uniformly at random.

ALGORITHM:

The algorithm works by generating an $$(n-2)$$-long random sequence of numbers chosen independently and uniformly from $$\{0,1,\ldots,n-1\}$$ and then applies an inverse Prufer transformation.

INPUT:

• n - number of vertices in the tree

EXAMPLES:

sage: G = graphs.RandomTree(10)
sage: G.is_tree()
True
sage: G.show() # long time

static RandomTreePowerlaw(n, gamma=3, tries=1000, seed=None)

Return a tree with a power law degree distribution, or False on failure.

From the NetworkX documentation: a trial power law degree sequence is chosen and then elements are swapped with new elements from a power law distribution until the sequence makes a tree (size = order - 1).

INPUT:

• n – number of vertices
• gamma – exponent of power law distribution
• tries – number of attempts to adjust sequence to make a tree
• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

We check that the generated graph is a tree:

sage: G = graphs.RandomTreePowerlaw(10, 3)
sage: G.is_tree()
True
sage: G.order(), G.size()
(10, 9)

sage: G = graphs.RandomTreePowerlaw(15, 2)
sage: if G:
....:     G.show()  # random output, long time

static RandomTriangulation(n, set_position=False)

Return a random triangulation on $$n$$ vertices.

A triangulation is a planar graph all of whose faces are triangles (3-cycles).

INPUT:

• $$n$$ – an integer
• set_position – boolean (default False) if set to True, this will compute coordinates for a planar drawing of the graph.

OUTPUT:

A random triangulation chosen uniformly among the rooted triangulations on $$n$$ vertices. This is a planar graph and comes with a combinatorial embedding.

Because some triangulations have nontrivial automorphism groups, this may not be equal to the uniform distribution among unrooted triangulations.

ALGORITHM:

The algorithm is taken from [PS2006], section 2.1.

Starting from a planar tree (represented by its contour as a sequence of vertices), one first performs local closures, until no one is possible. A local closure amounts to replace in the cyclic contour word a sequence in1,in2,in3,lf,in3 by in1,in3. After all local closures are done, one has reached the partial closure, as in [PS2006], figure 5 (a).

Then one has to perform complete closure by adding two more vertices, in order to reach the situation of [PS2006], figure 5 (b). For this, it is necessary to find inside the final contour one of the two subsequences lf,in,lf.

At every step of the algorithm, newly created edges are recorded in a graph, which will be returned at the end.

The combinatorial embedding is also computed and recorded in the output graph.

EXAMPLES:

sage: G = graphs.RandomTriangulation(6, True); G
Graph on 6 vertices
sage: G.is_planar()
True
sage: G.girth()
3
sage: G.plot(vertex_size=0, vertex_labels=False)
Graphics object consisting of 13 graphics primitives


REFERENCES:

 [PS2006] (1, 2, 3) Dominique Poulalhon and Gilles Schaeffer, Optimal coding and sampling of triangulations, Algorithmica 46 (2006), no. 3-4, 505-527, http://www.lix.polytechnique.fr/~poulalho/Articles/PoSc_Algorithmica06.pdf
static RingedTree(k, vertex_labels=True)

Return the ringed tree on k-levels.

A ringed tree of level $$k$$ is a binary tree with $$k$$ levels (counting the root as a level), in which all vertices at the same level are connected by a ring.

More precisely, in each layer of the binary tree (i.e. a layer is the set of vertices $$[2^i...2^{i+1}-1]$$) two vertices $$u,v$$ are adjacent if $$u=v+1$$ or if $$u=2^i$$ and $$v=2^{i+1}-1$$.

Ringed trees are defined in [CFHM12].

INPUT:

• k – the number of levels of the ringed tree.
• vertex_labels (boolean) – whether to label vertices as binary words (default) or as integers.

EXAMPLES:

sage: G = graphs.RingedTree(5)
sage: P = G.plot(vertex_labels=False, vertex_size=10)
sage: P.show() # long time
sage: G.vertices()
['', '0', '00', '000', '0000', '0001', '001', '0010', '0011', '01',
'010', '0100', '0101', '011', '0110', '0111', '1', '10', '100',
'1000', '1001', '101', '1010', '1011', '11', '110', '1100', '1101',
'111', '1110', '1111']


REFERENCES:

 [CFHM12] On the Hyperbolicity of Small-World and Tree-Like Random Graphs Wei Chen, Wenjie Fang, Guangda Hu, Michael W. Mahoney Arxiv 1201.1717
static RobertsonGraph()

Return the Robertson graph.

EXAMPLES:

sage: g = graphs.RobertsonGraph()
sage: g.order()
19
sage: g.size()
38
sage: g.diameter()
3
sage: g.girth()
5
sage: g.charpoly().factor()
(x - 4) * (x - 1)^2 * (x^2 + x - 5) * (x^2 + x - 1) * (x^2 - 3)^2 * (x^2 + x - 4)^2 * (x^2 + x - 3)^2
sage: g.chromatic_number()
3
sage: g.is_hamiltonian()
True
sage: g.is_vertex_transitive()
False

static RookGraph(dim_list, radius=None, relabel=False)

Returns the $$d$$-dimensional Rook’s Graph with prescribed dimensions.

The 2-dimensional Rook’s Graph of parameters $$n$$ and $$m$$ is a graph with $$nm$$ vertices in which each vertex represents a square in an $$n \times m$$ chessboard, and each edge corresponds to a legal move by a rook.

The $$d$$-dimensional Rook Graph with $$d >= 2$$ has for vertex set the cells of a $$d$$-dimensional grid with prescribed dimensions, and each edge corresponds to a legal move by a rook in any of the dimensions.

The Rook’s Graph for an $$n\times m$$ chessboard may also be defined as the Cartesian product of two complete graphs $$K_n \square K_m$$.

INPUT:

• dim_list – an iterable object (list, set, dict) providing the dimensions $$n_1, n_2, \ldots, n_d$$, with $$n_i \geq 1$$, of the chessboard.
• radius – (default: None) by setting the radius to a positive integer, one may decrease the power of the rook to at most radius steps. When the radius is 1, the resulting graph is a d-dimensional grid.
• relabel – (default: False) a boolean set to True if vertices must be relabeled as integers.

EXAMPLES:

The $$(n,m)$$-Rook’s Graph is isomorphic to the Cartesian product of two complete graphs:

sage: G = graphs.RookGraph( [3, 4] )
sage: H = ( graphs.CompleteGraph(3) ).cartesian_product( graphs.CompleteGraph(4) )
sage: G.is_isomorphic( H )
True


When the radius is 1, the Rook’s Graph is a grid:

sage: G = graphs.RookGraph( [3, 3, 4], radius=1 )
sage: H = graphs.GridGraph( [3, 4, 3] )
sage: G.is_isomorphic( H )
True

static SchlaefliGraph()

Return the Schläfli graph.

The Schläfli graph is the only strongly regular graphs of parameters $$(27,16,10,8)$$ (see [GR2001]).

Graph.is_strongly_regular() – tests whether a graph is strongly regular and/or returns its parameters.

Todo

Find a beautiful layout for this beautiful graph.

EXAMPLES:

Checking that the method actually returns the Schläfli graph:

sage: S = graphs.SchlaefliGraph()
sage: S.is_strongly_regular(parameters = True)
(27, 16, 10, 8)


The graph is vertex-transitive:

sage: S.is_vertex_transitive()
True


The neighborhood of each vertex is isomorphic to the complement of the Clebsch graph:

sage: neighborhood = S.subgraph(vertices = S.neighbors(0))
sage: graphs.ClebschGraph().complement().is_isomorphic(neighborhood)
True

static ShrikhandeGraph()

Return the Shrikhande graph.

For more information, see the MathWorld article on the Shrikhande graph or the Wikipedia article Shrikhande_graph.

Graph.is_strongly_regular() – tests whether a graph is strongly regular and/or returns its parameters.

EXAMPLES:

The Shrikhande graph was defined by S. S. Shrikhande in 1959. It has $$16$$ vertices and $$48$$ edges, and is strongly regular of degree $$6$$ with parameters $$(2,2)$$:

sage: G = graphs.ShrikhandeGraph(); G
Shrikhande graph: Graph on 16 vertices
sage: G.order()
16
sage: G.size()
48
sage: G.is_regular(6)
True
sage: set([ len([x for x in G.neighbors(i) if x in G.neighbors(j)])
....:     for i in range(G.order())
....:     for j in range(i) ])
{2}


It is non-planar, and both Hamiltonian and Eulerian:

sage: G.is_planar()
False
sage: G.is_hamiltonian()
True
sage: G.is_eulerian()
True


It has radius $$2$$, diameter $$2$$, and girth $$3$$:

sage: G.radius()
2
sage: G.diameter()
2
sage: G.girth()
3


Its chromatic number is $$4$$ and its automorphism group is of order $$192$$:

sage: G.chromatic_number()
4
sage: G.automorphism_group().cardinality()
192


It is an integral graph since it has only integral eigenvalues:

sage: G.characteristic_polynomial().factor()
(x - 6) * (x - 2)^6 * (x + 2)^9


It is a toroidal graph, and its embedding on a torus is dual to an embedding of the Dyck graph (DyckGraph).

static SierpinskiGasketGraph(n)

Return the Sierpinski Gasket graph of generation $$n$$.

All vertices but 3 have valence 4.

INPUT:

• $$n$$ – an integer

OUTPUT:

a graph $$S_n$$ with $$3 (3^{n-1}+1)/2$$ vertices and $$3^n$$ edges, closely related to the famous Sierpinski triangle fractal.

All these graphs have a triangular shape, and three special vertices at top, bottom left and bottom right. These are the only vertices of valence 2, all the other ones having valence 4.

The graph $$S_1$$ (generation $$1$$) is a triangle.

The graph $$S_{n+1}$$ is obtained from the disjoint union of three copies A,B,C of $$S_n$$ by identifying pairs of vertices: the top vertex of A with the bottom left vertex of B, the bottom right vertex of B with the top vertex of C, and the bottom left vertex of C with the bottom right vertex of A. There is another familly of graphs called Sierpinski graphs, where all vertices but 3 have valence 3. They are available using graphs.HanoiTowerGraph(3, n).

EXAMPLES:

sage: s4 = graphs.SierpinskiGasketGraph(4); s4
Graph on 42 vertices
sage: s4.size()
81
sage: s4.degree_histogram()
[0, 0, 3, 0, 39]
sage: s4.is_hamiltonian()
True


REFERENCES:

 [LLWC] Chien-Hung Lin, Jia-Jie Liu, Yue-Li Wang, William Chung-Kung Yen, The Hub Number of Sierpinski-Like Graphs, Theory Comput Syst (2011), vol 49, doi:10.1007/s00224-010-9286-3
static SimsGewirtzGraph()

Return the Sims-Gewirtz Graph.

This graph is obtained from the Higman Sims graph by considering the graph induced by the vertices at distance two from the vertices of an (any) edge. It is the only strongly regular graph with parameters $$v = 56$$, $$k = 10$$, $$\lambda = 0$$, $$\mu = 2$$

For more information on the Sylvester graph, see https://www.win.tue.nl/~aeb/graphs/Sims-Gewirtz.html or its Wikipedia article Gewirtz_graph.

EXAMPLES:

sage: g = graphs.SimsGewirtzGraph(); g
Sims-Gewirtz Graph: Graph on 56 vertices
sage: g.order()
56
sage: g.size()
280
sage: g.is_strongly_regular(parameters = True)
(56, 10, 0, 2)

static SousselierGraph()

Return the Sousselier Graph.

The Sousselier graph is a hypohamiltonian graph on 16 vertices and 27 edges. For more information, see Wikipedia article Sousselier_graph or the corresponding French Wikipedia page.

EXAMPLES:

sage: g = graphs.SousselierGraph()
sage: g.order()
16
sage: g.size()
27
2
sage: g.diameter()
3
sage: g.automorphism_group().cardinality()
2
sage: g.is_hamiltonian()
False
sage: g.delete_vertex(g.random_vertex())
sage: g.is_hamiltonian()
True

static SquaredSkewHadamardMatrixGraph(n)

Pseudo-$$OA(2n,4n-1)$$-graph from a skew Hadamard matrix of order $$4n$$

A strongly regular graph with parameters of the orthogonal array graph OrthogonalArrayBlockGraph, also known as pseudo Latin squares graph $$L_{2n}(4n-1)$$, constructed from a skew Hadamard matrix of order $$4n$$, due to Goethals and Seidel, see [BvL84].

EXAMPLES:

sage: graphs.SquaredSkewHadamardMatrixGraph(4).is_strongly_regular(parameters=True)
(225, 112, 55, 56)
(1225, 612, 305, 306)

static StarGraph(n)

Return a star graph with $$n+1$$ nodes.

A Star graph is a basic structure where one node is connected to all other nodes.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each star graph will be displayed with the first (0) node in the center, the second node (1) at the top, with the rest following in a counterclockwise manner. (0) is the node connected to all other nodes.

The star graph is a good opportunity to compare efficiency of filling a position dictionary vs. using the spring-layout algorithm for plotting. As far as display, the spring-layout should push all other nodes away from the (0) node, and thus look very similar to this constructor’s positioning.

EXAMPLES:

sage: import networkx


Compare the plots:

sage: n = networkx.star_graph(23)
sage: spring23 = Graph(n)
sage: posdict23 = graphs.StarGraph(23)
sage: spring23.show() # long time
sage: posdict23.show() # long time


View many star graphs as a Sage Graphics Array

With this constructor (i.e., the position dictionary filled)

sage: g = []
sage: j = []
sage: for i in range(9):
....:     k = graphs.StarGraph(i+3)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


Compared to plotting with the spring-layout algorithm

sage: g = []
sage: j = []
sage: for i in range(9):
....:     spr = networkx.star_graph(i+3)
....:     k = Graph(spr)
....:     g.append(k)
sage: for i in range(3):
....:     n = []
....:     for m in range(3):
....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static SuzukiGraph()

Return the Suzuki Graph.

The Suzuki graph has 1782 vertices, and is strongly regular with parameters $$(1782,416,100,96)$$. Known as S.15 in [Hu75].

Note

It takes approximately 50 seconds to build this graph. Do not be too impatient.

EXAMPLES:

sage: g = graphs.SuzukiGraph(); g            # optional internet # not tested
Suzuki graph: Graph on 1782 vertices
sage: g.is_strongly_regular(parameters=True) # optional internet # not tested
(1782, 416, 100, 96)

static SwitchedSquaredSkewHadamardMatrixGraph(n)

A strongly regular graph in Seidel switching class of $$SquaredSkewHadamardMatrixGraph$$

A strongly regular graph in the Seidel switching class of the disjoint union of a 1-vertex graph and the one produced by Pseudo-L_{2n}(4n-1)

In this case, the other possible parameter set of a strongly regular graph in the Seidel switching class of the latter graph (see [BH12]) coincides with the set of parameters of the complement of the graph returned by this function.

EXAMPLES:

sage: g=graphs.SwitchedSquaredSkewHadamardMatrixGraph(4)
sage: g.is_strongly_regular(parameters=True)
(226, 105, 48, 49)
sage: from sage.combinat.designs.twographs import twograph_descendant
sage: twograph_descendant(g,0).is_strongly_regular(parameters=True)
(225, 112, 55, 56)
sage: twograph_descendant(g.complement(),0).is_strongly_regular(parameters=True)
(225, 112, 55, 56)

static SylvesterGraph()

Return the Sylvester Graph.

This graph is obtained from the Hoffman Singleton graph by considering the graph induced by the vertices at distance two from the vertices of an (any) edge.

EXAMPLES:

sage: g = graphs.SylvesterGraph(); g
Sylvester Graph: Graph on 36 vertices
sage: g.order()
36
sage: g.size()
90
sage: g.is_regular(k=5)
True

static SymplecticDualPolarGraph(m, q)

Returns the Symplectic Dual Polar Graph $$DSp(m,q)$$.

For more information on Symplectic Dual Polar graphs, see [BCN1989] and Sect. 2.3.1 of [Co81].

INPUT:

• m,q (integers) – $$q$$ must be a prime power, and $$m$$ must be even.

EXAMPLES:

sage: G = graphs.SymplecticDualPolarGraph(6,3); G       # not tested (long time)
Symplectic Dual Polar Graph DSp(6, 3): Graph on 1120 vertices
sage: G.is_distance_regular(parameters=True)            # not tested (long time)
([39, 36, 27, None], [None, 1, 4, 13])


REFERENCE:

 [Co81] (1, 2) A. M. Cohen, A synopsis of known distance-regular graphs with large diameters, Stichting Mathematisch Centrum, 1981.
static SymplecticPolarGraph(d, q, algorithm=None)

Returns the Symplectic Polar Graph $$Sp(d,q)$$.

The Symplectic Polar Graph $$Sp(d,q)$$ is built from a projective space of dimension $$d-1$$ over a field $$F_q$$, and a symplectic form $$f$$. Two vertices $$u,v$$ are made adjacent if $$f(u,v)=0$$.

See the page on symplectic graphs on Andries Brouwer’s website.

INPUT:

• d,q (integers) – note that only even values of $$d$$ are accepted by the function.
• algorithm – if set to ‘gap’ then the computation is carried via GAP library interface, computing totally singular subspaces, which is faster for $$q>3$$. Otherwise it is done directly.

EXAMPLES:

Computation of the spectrum of $$Sp(6,2)$$:

sage: g = graphs.SymplecticPolarGraph(6,2)
sage: g.is_strongly_regular(parameters=True)
(63, 30, 13, 15)
sage: set(g.spectrum()) == {-5, 3, 30}
True


The parameters of $$Sp(4,q)$$ are the same as of $$O(5,q)$$, but they are not isomorphic if $$q$$ is odd:

sage: G = graphs.SymplecticPolarGraph(4,3)
sage: G.is_strongly_regular(parameters=True)
(40, 12, 2, 4)
sage: O=graphs.OrthogonalPolarGraph(5,3)
sage: O.is_strongly_regular(parameters=True)
(40, 12, 2, 4)
sage: O.is_isomorphic(G)
False
sage: graphs.SymplecticPolarGraph(6,4,algorithm="gap").is_strongly_regular(parameters=True) # not tested (long time)
(1365, 340, 83, 85)

static SzekeresSnarkGraph()

Return the Szekeres Snark Graph.

The Szekeres graph is a snark with 50 vertices and 75 edges. For more information on this graph, see the Wikipedia article Szekeres_snark.

EXAMPLES:

sage: g = graphs.SzekeresSnarkGraph()
sage: g.order()
50
sage: g.size()
75
sage: g.chromatic_number()
3

static T2starGeneralizedQuadrangleGraph(q, dual=False, hyperoval=None, field=None, check_hyperoval=True)

Return the collinearity graph of the generalized quadrangle $$T_2^*(q)$$, or of its dual

Let $$q=2^k$$ and $$\Theta=PG(3,q)$$. $$T_2^*(q)$$ is a generalized quadrangle [GQwiki] of order $$(q-1,q+1)$$, see 3.1.3 in [PT09]. Fix a plane $$\Pi \subset \Theta$$ and a hyperoval $$O \subset \Pi$$. The points of $$T_2^*(q):=T_2^*(O)$$ are the points of $$\Theta$$ outside $$\Pi$$, and the lines are the lines of $$\Theta$$ outside $$\Pi$$ that meet $$\Pi$$ in a point of $$O$$.

INPUT:

• q – a power of two
• dual – if False (default), return the graph of $$T_2^*(O)$$. Otherwise return the graph of the dual $$T_2^*(O)$$.
• hyperoval – a hyperoval (i.e. a complete 2-arc; a set of points in the plane meeting every line in 0 or 2 points) in the plane of points with 0th coordinate 0 in $$PG(3,q)$$ over the field field. Each point of hyperoval must be a length 4 vector over field with 1st non-0 coordinate equal to 1. By default, hyperoval and field are not specified, and constructed on the fly. In particular, hyperoval we build is the classical one, i.e. a conic with the point of intersection of its tangent lines.
• field – an instance of a finite field of order $$q$$, must be provided if hyperoval is provided.
• check_hyperoval – (default: True) if True, check hyperoval for correctness.

EXAMPLES:

using the built-in construction:

sage: g=graphs.T2starGeneralizedQuadrangleGraph(4); g
T2*(O,4); GQ(3, 5): Graph on 64 vertices
sage: g.is_strongly_regular(parameters=True)
(64, 18, 2, 6)
T2*(O,4)*; GQ(5, 3): Graph on 96 vertices
sage: g.is_strongly_regular(parameters=True)
(96, 20, 4, 4)


sage: F=GF(4,'b')
sage: O=[vector(F,(0,0,0,1)),vector(F,(0,0,1,0))]+[vector(F, (0,1,x^2,x)) for x in F]
T2*(O,4); GQ(3, 5): Graph on 64 vertices
sage: g.is_strongly_regular(parameters=True)
(64, 18, 2, 6)

static TadpoleGraph(n1, n2)

Return a tadpole graph with n1+n2 nodes.

A tadpole graph is a path graph (order n2) connected to a cycle graph (order n1).

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, the cycle graph will be drawn in the lower-left corner with the (n1)th node at a 45 degree angle above the right horizontal center of the cycle graph, leading directly into the path graph.

EXAMPLES:

Construct and show a tadpole graph Cycle = 13, Stick = 4:

sage: g = graphs.TadpoleGraph(13, 4); g
Tadpole graph: Graph on 17 vertices
sage: g.show() # long time

static TaylorTwographDescendantSRG(q, clique_partition=None)

constructing the descendant graph of the Taylor’s two-graph for $$U_3(q)$$, $$q$$ odd

This is a strongly regular graph with parameters $$(v,k,\lambda,\mu)=(q^3, (q^2+1)(q-1)/2, (q-1)^3/4-1, (q^2+1)(q-1)/4)$$ obtained as a two-graph descendant of the Taylor's two-graph $$T$$. This graph admits a partition into cliques of size $$q$$, which are useful in TaylorTwographSRG(), a strongly regular graph on $$q^3+1$$ vertices in the Seidel switching class of $$T$$, for which we need $$(q^2+1)/2$$ cliques. The cliques are the $$q^2$$ lines on $$v_0$$ of the projective plane containing the unital for $$U_3(q)$$, and intersecting the unital (i.e. the vertices of the graph and the point we remove) in $$q+1$$ points. This is all taken from §7E of [BvL84].

INPUT:

• q – a power of an odd prime number
• clique_partition – if True, return $$q^2-1$$ cliques of size $$q$$ with empty pairwise intersection. (Removing all of them leaves a clique, too), and the point removed from the unital.

EXAMPLES:

sage: g=graphs.TaylorTwographDescendantSRG(3); g
Taylor two-graph descendant SRG: Graph on 27 vertices
sage: g.is_strongly_regular(parameters=True)
(27, 10, 1, 5)
sage: from sage.combinat.designs.twographs import taylor_twograph
sage: T = taylor_twograph(3)                           # long time
sage: g.is_isomorphic(T.descendant(T.ground_set())) # long time
True
sage: g=graphs.TaylorTwographDescendantSRG(5)    # not tested (long time)
sage: g.is_strongly_regular(parameters=True)  # not tested (long time)
(125, 52, 15, 26)

static TaylorTwographSRG(q)

constructing a strongly regular graph from the Taylor’s two-graph for $$U_3(q)$$, $$q$$ odd

This is a strongly regular graph with parameters $$(v,k,\lambda,\mu)=(q^3+1, q(q^2+1)/2, (q^2+3)(q-1)/4, (q^2+1)(q+1)/4)$$ in the Seidel switching class of Taylor two-graph. Details are in §7E of [BvL84].

INPUT:

• q – a power of an odd prime number

EXAMPLES:

sage: t=graphs.TaylorTwographSRG(3); t
Taylor two-graph SRG: Graph on 28 vertices
sage: t.is_strongly_regular(parameters=True)
(28, 15, 6, 10)

static TetrahedralGraph()

Returns a tetrahedral graph (with 4 nodes).

A tetrahedron is a 4-sided triangular pyramid. The tetrahedral graph corresponds to the connectivity of the vertices of the tetrahedron. This graph is equivalent to a wheel graph with 4 nodes and also a complete graph on four nodes. (See examples below).

PLOTTING: The tetrahedral graph should be viewed in 3 dimensions. We chose to use the default spring-layout algorithm here, so that multiple iterations might yield a different point of reference for the user. We hope to add rotatable, 3-dimensional viewing in the future. In such a case, a string argument will be added to select the flat spring-layout over a future implementation.

EXAMPLES: Construct and show a Tetrahedral graph

sage: g = graphs.TetrahedralGraph()
sage: g.show() # long time


The following example requires networkx:

sage: import networkx as NX


Compare this Tetrahedral, Wheel(4), Complete(4), and the Tetrahedral plotted with the spring-layout algorithm below in a Sage graphics array:

sage: tetra_pos = graphs.TetrahedralGraph()
sage: tetra_spring = Graph(NX.tetrahedral_graph())
sage: wheel = graphs.WheelGraph(4)
sage: complete = graphs.CompleteGraph(4)
sage: g = [tetra_pos, tetra_spring, wheel, complete]
sage: j = []
sage: for i in range(2):
....:     n = []
....:     for m in range(2):
....:         n.append(g[i + m].plot(vertex_size=50, vertex_labels=False))
....:     j.append(n)
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time

static ThomsenGraph()

Return the Thomsen Graph.

The Thomsen Graph is actually a complete bipartite graph with $$(n1, n2) = (3, 3)$$. It is also called the Utility graph.

PLOTTING: See CompleteBipartiteGraph.

EXAMPLES:

sage: T = graphs.ThomsenGraph()
sage: T
Thomsen graph: Graph on 6 vertices
sage: T.graph6_string()
'EFz_'
sage: (graphs.ThomsenGraph()).show() # long time

static TietzeGraph()

Return the Tietze Graph.

For more information on the Tietze Graph, see the Wikipedia article Tietze’s_graph.

EXAMPLES:

sage: g = graphs.TietzeGraph()
sage: g.order()
12
sage: g.size()
18
sage: g.diameter()
3
sage: g.girth()
3
sage: g.automorphism_group().cardinality()
12
sage: g.automorphism_group().is_isomorphic(groups.permutation.Dihedral(6))
True

static ToleranceGraph(tolrep)

Returns the graph generated by the tolerance representation tolrep.

The tolerance representation tolrep is described by the list $$((l_0,r_0,t_0), (l_1,r_1,t_1), ..., (l_k,r_k,t_k))$$ where $$I_i = (l_i,r_i)$$ denotes a closed interval on the real line with $$l_i < r_i$$ and $$t_i$$ a positive value, called tolerance. This representation generates the tolerance graph with the vertex set {0,1, …, k} and the edge set $${(i,j): |I_i \cap I_j| \ge \min{t_i, t_j}}$$ where $$|I_i \cap I_j|$$ denotes the length of the intersection of $$I_i$$ and $$I_j$$.

INPUT:

• tolrep – list of triples $$(l_i,r_i,t_i)$$ where $$(l_i,r_i)$$ denotes a closed interval on the real line and $$t_i$$ a positive value.

Note

The vertices are named 0, 1, …, k. The tolerance representation used to create the graph is saved with the graph and can be recovered using get_vertex() or get_vertices().

EXAMPLES:

The following code creates a tolerance representation tolrep, generates its tolerance graph g, and applies some checks:

sage: tolrep = [(1,4,3),(1,2,1),(2,3,1),(0,3,3)]
sage: g = graphs.ToleranceGraph(tolrep)
sage: g.get_vertex(3)
(0, 3, 3)
sage: neigh = g.neighbors(3)
sage: for v in neigh: print(g.get_vertex(v))
(1, 2, 1)
(2, 3, 1)
sage: g.is_interval()
False
sage: g.is_weakly_chordal()
True


The intervals in the list need not be distinct

sage: tolrep2 = [(0,4,5),(1,2,1),(2,3,1),(0,4,5)]
sage: g2 = graphs.ToleranceGraph(tolrep2)
sage: g2.get_vertices()
{0: (0, 4, 5), 1: (1, 2, 1), 2: (2, 3, 1), 3: (0, 4, 5)}
sage: g2.is_isomorphic(g)
True


Real values are also allowed

sage: tolrep = [(0.1,3.3,4.4),(1.1,2.5,1.1),(1.4,4.4,3.3)]
sage: g = graphs.ToleranceGraph(tolrep)
sage: g.is_isomorphic(graphs.PathGraph(3))
True

static Toroidal6RegularGrid2dGraph(n1, n2)

Returns a toroidal 6-regular grid.

The toroidal 6-regular grid is a 6-regular graph on $$n_1\times n_2$$ vertices and its elements have coordinates $$(i,j)$$ for $$i \in \{0...i-1\}$$ and $$j \in \{0...j-1\}$$.

Its edges are those of the ToroidalGrid2dGraph(), to which are added the edges between $$(i,j)$$ and $$((i+1)\%n_1, (j+1)\%n_2)$$.

INPUT:

• n1, n2 (integers) – see above.

EXAMPLES:

The toroidal 6-regular grid on $$25$$ elements:

sage: g = graphs.Toroidal6RegularGrid2dGraph(5,5)
sage: g.is_regular(k=6)
True
sage: g.is_vertex_transitive()
True
sage: g.line_graph().is_vertex_transitive()
True
sage: g.automorphism_group().cardinality()
300
sage: g.is_hamiltonian()
True

static ToroidalGrid2dGraph(n1, n2)

Returns a toroidal 2-dimensional grid graph with $$n_1n_2$$ nodes ($$n_1$$ rows and $$n_2$$ columns).

The toroidal 2-dimensional grid with parameters $$n_1,n_2$$ is the 2-dimensional grid graph with identical parameters to which are added the edges $$((i,0),(i,n_2-1))$$ and $$((0,i),(n_1-1,i))$$.

EXAMPLES:

The toroidal 2-dimensional grid is a regular graph, while the usual 2-dimensional grid is not

sage: tgrid = graphs.ToroidalGrid2dGraph(8,9)
sage: print(tgrid)
Toroidal 2D Grid Graph with parameters 8,9
sage: grid = graphs.Grid2dGraph(8,9)
sage: grid.is_regular()
False
sage: tgrid.is_regular()
True

static TruncatedIcosidodecahedralGraph()

Return the truncated icosidodecahedron.

The truncated icosidodecahedron is an Archimedean solid with 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges. For more information, see the Wikipedia article Truncated_icosidodecahedron.

EXAMPLES:

Unfortunately, this graph can not be constructed currently, due to numerical issues:

sage: g = graphs.TruncatedIcosidodecahedralGraph(); g
Traceback (most recent call last):
...
ValueError: *Error: Numerical inconsistency is found.  Use the GMP exact arithmetic.
sage: g.order(), g.size() # not tested
(120, 180)

static TruncatedTetrahedralGraph()

Return the truncated tetrahedron.

The truncated tetrahedron is an Archimedean solid with 12 vertices and 18 edges. For more information, see the Wikipedia article Truncated_tetrahedron.

EXAMPLES:

sage: g = graphs.TruncatedTetrahedralGraph(); g
Truncated Tetrahedron: Graph on 12 vertices
sage: g.order(), g.size()
(12, 18)
sage: g.is_isomorphic(polytopes.simplex(3).truncation().graph())
True

static TuranGraph(n, r)

Returns the Turan graph with parameters $$n, r$$.

Turan graphs are complete multipartite graphs with $$n$$ vertices and $$r$$ subsets, denoted $$T(n,r)$$, with the property that the sizes of the subsets are as close to equal as possible. The graph $$T(n,r)$$ will have $$n \pmod r$$ subsets of size $$\lfloor n/r \rfloor$$ and $$r - (n \pmod r)$$ subsets of size $$\lceil n/r \rceil$$. See the Wikipedia article Turan_graph for more information.

INPUT:

• n (integer)– the number of vertices in the graph.
• r (integer) – the number of partitions of the graph.

EXAMPLES:

The Turan graph is a complete multipartite graph.

sage: g = graphs.TuranGraph(13, 4)
sage: k = graphs.CompleteMultipartiteGraph([3,3,3,4])
sage: g.is_isomorphic(k)
True


The Turan graph $$T(n,r)$$ has $$\lfloor \frac{(r-1)(n^2)}{2r} \rfloor$$ edges.

sage: n = 13
sage: r = 4
sage: g = graphs.TuranGraph(n,r)
sage: g.size() == floor((r-1)*(n**2)/(2*r))
True

static Tutte12Cage()

Return the Tutte 12-Cage.

See the Wikipedia article Tutte_12-cage.

EXAMPLES:

sage: g = graphs.Tutte12Cage()
sage: g.order()
126
sage: g.size()
189
sage: g.girth()
12
sage: g.diameter()
6
sage: g.show()

static TutteCoxeterGraph(embedding=2)

Return the Tutte-Coxeter graph.

INPUT:

• embedding – two embeddings are available, and can be selected by setting embedding to 1 or 2.

EXAMPLES:

sage: g = graphs.TutteCoxeterGraph()
sage: g.order()
30
sage: g.size()
45
sage: g.girth()
8
sage: g.diameter()
4
sage: g.show()
sage: graphs.TutteCoxeterGraph(embedding=1).show()

static TutteGraph()

Return the Tutte Graph.

The Tutte graph is a 3-regular, 3-connected, and planar non-hamiltonian graph. For more information on the Tutte Graph, see the Wikipedia article Tutte_graph.

EXAMPLES:

sage: g = graphs.TutteGraph()
sage: g.order()
46
sage: g.size()
69
sage: g.is_planar()
True
sage: g.vertex_connectivity() # long time
3
sage: g.girth()
4
sage: g.automorphism_group().cardinality()
3
sage: g.is_hamiltonian()
False

static U42Graph216()

Return a (216,40,4,8)-strongly regular graph from [CRS2016].

Build the graph, interpreting the $$U_4(2)$$-action considered in [CRS2016] as the one on the hyperbolic lines of the corresponding unitary polar space, and then doing the unique merging of the orbitals leading to a graph with the parameters in question.

EXAMPLES:

sage: G=graphs.U42Graph216()                 # optional - gap_packages (grape)
sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape)
(216, 40, 4, 8)

static U42Graph540()

Return a (540,187,58,68)-strongly regular graph from [CRS2016].

Build the graph, interpreting the $$U_4(2)$$-action considered in [CRS2016] as the action of $$U_4(2)=Sp_4(3)<U_4(3)$$ on the nonsingular, w.r.t. to the Hermitean form stabilised by $$U_4(3)$$, points of the 3-dimensional projective space over $$GF(9)$$. There are several possible mergings of orbitals, some leading to non-isomorphic graphs with the same parameters. We found the merging here using [COCO].

EXAMPLES:

sage: G=graphs.U42Graph540() # optional - gap_packages (grape)
sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape)
(540, 187, 58, 68)

static USAMap(continental=False)

Return states of USA as a graph of common border.

The graph has an edge between those states that have common land border line or point. Hence for example Colorado and Arizona are marked as neighbors, but Michigan and Minnesota are not.

INPUT:

• continental, a Boolean – if set, exclude Alaska and Hawaii

EXAMPLES:

How many states are neighbor’s neighbor for Pennsylvania:

sage: USA = graphs.USAMap()
sage: len([n2 for n2 in USA if USA.distance('Pennsylvania', n2) == 2])
7


Diameter for continental USA:

sage: USAcont = graphs.USAMap(continental=True)
sage: USAcont.diameter()
11

static UnitaryDualPolarGraph(m, q)

Returns the Dual Unitary Polar Graph $$U(m,q)$$.

For more information on Unitary Dual Polar graphs, see [BCN1989] and Sect. 2.3.1 of [Co81].

INPUT:

• m,q (integers) – $$q$$ must be a prime power.

EXAMPLES:

The point graph of a generalized quadrangle (see [GQwiki], [PT09]) of order (8,4):

sage: G = graphs.UnitaryDualPolarGraph(5,2); G   # long time
Unitary Dual Polar Graph DU(5, 2); GQ(8, 4): Graph on 297 vertices
sage: G.is_strongly_regular(parameters=True)     # long time
(297, 40, 7, 5)


Another way to get the generalized quadrangle of order (2,4):

sage: G = graphs.UnitaryDualPolarGraph(4,2); G
Unitary Dual Polar Graph DU(4, 2); GQ(2, 4): Graph on 27 vertices
sage: G.is_isomorphic(graphs.OrthogonalPolarGraph(6,2,'-'))
True


A bigger graph:

sage: G = graphs.UnitaryDualPolarGraph(6,2); G   # not tested (long time)
Unitary Dual Polar Graph DU(6, 2): Graph on 891 vertices
sage: G.is_distance_regular(parameters=True)     # not tested (long time)
([42, 40, 32, None], [None, 1, 5, 21])

static UnitaryPolarGraph(m, q, algorithm='gap')

Returns the Unitary Polar Graph $$U(m,q)$$.

For more information on Unitary Polar graphs, see the page of Andries Brouwer’s website.

INPUT:

• m,q (integers) – $$q$$ must be a prime power.
• algorithm – if set to ‘gap’ then the computation is carried via GAP library interface, computing totally singular subspaces, which is faster for large examples (especially with $$q>2$$). Otherwise it is done directly.

EXAMPLES:

sage: G = graphs.UnitaryPolarGraph(4,2); G
Unitary Polar Graph U(4, 2); GQ(4, 2): Graph on 45 vertices
sage: G.is_strongly_regular(parameters=True)
(45, 12, 3, 3)
sage: graphs.UnitaryPolarGraph(5,2).is_strongly_regular(parameters=True)
(165, 36, 3, 9)
sage: graphs.UnitaryPolarGraph(6,2)    # not tested (long time)
Unitary Polar Graph U(6, 2): Graph on 693 vertices

static WagnerGraph()

Return the Wagner Graph.

See the Wikipedia article Wagner_graph.

EXAMPLES:

sage: g = graphs.WagnerGraph()
sage: g.order()
8
sage: g.size()
12
sage: g.girth()
4
sage: g.diameter()
2
sage: g.show()

static WatkinsSnarkGraph()

Return the Watkins Snark Graph.

The Watkins Graph is a snark with 50 vertices and 75 edges. For more information, see the Wikipedia article Watkins_snark.

EXAMPLES:

sage: g = graphs.WatkinsSnarkGraph()
sage: g.order()
50
sage: g.size()
75
sage: g.chromatic_number()
3

static WellsGraph()

Return the Wells graph.

The implementation follows the construction given on page 266 of [BCN1989]. This requires to create intermediate graphs and run a small isomorphism test, while everything could be replaced by a pre-computed list of edges : I believe that it is better to keep “the recipe” in the code, however, as it is quite unlikely that this could become the most time-consuming operation in any sensible algorithm, and …. “preserves knowledge”, which is what open-source software is meant to do.

EXAMPLES:

sage: g = graphs.WellsGraph(); g
Wells graph: Graph on 32 vertices
sage: g.order()
32
sage: g.size()
80
sage: g.girth()
5
sage: g.diameter()
4
sage: g.chromatic_number()
4
sage: g.is_regular(k=5)
True

static WheelGraph(n)

Returns a Wheel graph with n nodes.

A Wheel graph is a basic structure where one node is connected to all other nodes and those (outer) nodes are connected cyclically.

PLOTTING: Upon construction, the position dictionary is filled to override the spring-layout algorithm. By convention, each wheel graph will be displayed with the first (0) node in the center, the second node at the top, and the rest following in a counterclockwise manner.

With the wheel graph, we see that it doesn’t take a very large n at all for the spring-layout to give a counter-intuitive display. (See Graphics Array examples below).

EXAMPLES:

We view many wheel graphs with a Sage Graphics Array, first with this constructor (i.e., the position dictionary filled):

sage: g = []
sage: j = []
sage: for i in range(9):
....:  k = graphs.WheelGraph(i+3)
....:  g.append(k)
...
sage: for i in range(3):
....:  n = []
....:  for m in range(3):
....:      n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:  j.append(n)
...
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


Next, using the spring-layout algorithm:

sage: import networkx
sage: g = []
sage: j = []
sage: for i in range(9):
....:  spr = networkx.wheel_graph(i+3)
....:  k = Graph(spr)
....:  g.append(k)
...
sage: for i in range(3):
....:  n = []
....:  for m in range(3):
....:      n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
....:  j.append(n)
...
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time


Compare the plotting:

sage: n = networkx.wheel_graph(23)
sage: spring23 = Graph(n)
sage: posdict23 = graphs.WheelGraph(23)
sage: spring23.show() # long time
sage: posdict23.show() # long time

static WienerArayaGraph()

Return the Wiener-Araya Graph.

The Wiener-Araya Graph is a planar hypohamiltonian graph on 42 vertices and 67 edges. For more information, see the Wolfram Page on the Wiener-Araya Graph or Wikipedia article Wiener-Araya_graph.

EXAMPLES:

sage: g = graphs.WienerArayaGraph()
sage: g.order()
42
sage: g.size()
67
sage: g.girth()
4
sage: g.is_planar()
True
sage: g.is_hamiltonian() # not tested -- around 30s long
False
sage: g.delete_vertex(g.random_vertex())
sage: g.is_hamiltonian()
True

static WindmillGraph(k, n)

Return the Windmill graph $$Wd(k, n)$$.

The windmill graph $$Wd(k, n)$$ is an undirected graph constructed for $$k \geq 2$$ and $$n \geq 2$$ by joining $$n$$ copies of the complete graph $$K_k$$ at a shared vertex. It has $$(k-1)n+1$$ vertices and $$nk(k-1)/2$$ edges, girth 3 (if $$k > 2$$), radius 1 and diameter 2. It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is $$(k-1)$$-edge-connected. It is trivially perfect and a block graph.

EXAMPLES:

The Windmill graph $$Wd(2, n)$$ is a star graph:

sage: n = 5
sage: W = graphs.WindmillGraph(2, n)
sage: W.is_isomorphic( graphs.StarGraph(n) )
True


The Windmill graph $$Wd(3, n)$$ is the Friendship graph $$F_n$$:

sage: n = 5
sage: W = graphs.WindmillGraph(3, n)
sage: W.is_isomorphic( graphs.FriendshipGraph(n) )
True


The Windmill graph $$Wd(3, 2)$$ is the Butterfly graph:

sage: W = graphs.WindmillGraph(3, 2)
sage: W.is_isomorphic( graphs.ButterflyGraph() )
True


The Windmill graph $$Wd(k, n)$$ has chromatic number $$k$$:

sage: n,k = 5,6
sage: W = graphs.WindmillGraph(k, n)
sage: W.chromatic_number() == k
True

static WorldMap()

Returns the Graph of all the countries, in which two countries are adjacent in the graph if they have a common boundary.

This graph has been built from the data available in The CIA World Factbook [CIA] (2009-08-21).

The returned graph G has a member G.gps_coordinates equal to a dictionary containing the GPS coordinates of each country’s capital city.

EXAMPLES:

sage: g = graphs.WorldMap()
sage: g.has_edge("France", "Italy")
True
sage: g.gps_coordinates["Bolivia"]
[[17, 'S'], [65, 'W']]
sage: sorted(g.connected_component_containing_vertex('Ireland'))
['Ireland', 'United Kingdom']


REFERENCE:

static chang_graphs()

Return the three Chang graphs.

Three of the four strongly regular graphs of parameters $$(28,12,6,4)$$ are called the Chang graphs. The fourth is the line graph of $$K_8$$. For more information about the Chang graphs, see the Wikipedia article Chang_graphs or https://www.win.tue.nl/~aeb/graphs/Chang.html.

EXAMPLES: check that we get 4 non-isomorphic s.r.g.’s with the same parameters:

sage: chang_graphs = graphs.chang_graphs()
sage: K8 = graphs.CompleteGraph(8)
sage: T8 = K8.line_graph()
sage: four_srg = chang_graphs + [T8]
sage: for g in four_srg:
....:     print(g.is_strongly_regular(parameters=True))
(28, 12, 6, 4)
(28, 12, 6, 4)
(28, 12, 6, 4)
(28, 12, 6, 4)
sage: from itertools import combinations
sage: for g1,g2 in combinations(four_srg,2):
....:     assert not g1.is_isomorphic(g2)


Construct the Chang graphs by Seidel switching:

sage: c3c5=graphs.CycleGraph(3).disjoint_union(graphs.CycleGraph(5))
sage: c8=graphs.CycleGraph(8)
sage: s=[K8.subgraph_search(c8).edges(),
....:    [(0,1,None),(2,3,None),(4,5,None),(6,7,None)],
....:    K8.subgraph_search(c3c5).edges()]
sage: list(map(lambda x,G: T8.seidel_switching(x, inplace=False).is_isomorphic(G),
....:                  s, chang_graphs))
[True, True, True]

cospectral_graphs(vertices, matrix_function=<function <lambda>>, graphs=None)

Find all sets of graphs on vertices vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.

INPUT:

• vertices - The number of vertices in the graphs to be tested

• matrix_function - A function taking a graph and giving back a matrix. This defaults to the adjacency matrix. The spectra examined are the spectra of these matrices.

• graphs - One of three things:

• None (default) - test all graphs having vertices vertices
• a function taking a graph and returning True or False - test only the graphs on vertices vertices for which the function returns True
• a list of graphs (or other iterable object) - these graphs are tested for cospectral sets. In this case, vertices is ignored.

OUTPUT:

A list of lists of graphs. Each sublist will be a list of cospectral graphs (lists of cardinality 1 being omitted).

Graph.is_strongly_regular() – tests whether a graph is strongly regular and/or returns its parameters.

EXAMPLES:

sage: g=graphs.cospectral_graphs(5)
sage: sorted(sorted(g.graph6_string() for g in glist) for glist in g)
[['Dr?', 'Ds_']]
sage: g.am().charpoly()==g.am().charpoly()
True


There are two sets of cospectral graphs on six vertices with no isolated vertices:

sage: g=graphs.cospectral_graphs(6, graphs=lambda x: min(x.degree())>0)
sage: sorted(sorted(g.graph6_string() for g in glist) for glist in g)
[['Ep__', 'Er?G'], ['ExGg', 'ExoG']]
sage: g.am().charpoly()==g.am().charpoly()
True
sage: g.am().charpoly()==g.am().charpoly()
True


There is one pair of cospectral trees on eight vertices:

sage: g=graphs.cospectral_graphs(6, graphs=graphs.trees(8))
sage: sorted(sorted(g.graph6_string() for g in glist) for glist in g)
[['GiPC?C', 'GiQCC?']]
sage: g.am().charpoly()==g.am().charpoly()
True


There are two sets of cospectral graphs (with respect to the Laplacian matrix) on six vertices:

sage: g=graphs.cospectral_graphs(6, matrix_function=lambda g: g.laplacian_matrix())
sage: sorted(sorted(g.graph6_string() for g in glist) for glist in g)
[['Edq_', 'ErcG'], ['Exoo', 'EzcG']]
sage: g.laplacian_matrix().charpoly()==g.laplacian_matrix().charpoly()
True
sage: g.laplacian_matrix().charpoly()==g.laplacian_matrix().charpoly()
True


To find cospectral graphs with respect to the normalized Laplacian, assuming the graphs do not have an isolated vertex, it is enough to check the spectrum of the matrix $$D^{-1}A$$, where $$D$$ is the diagonal matrix of vertex degrees, and A is the adjacency matrix. We find two such cospectral graphs (for the normalized Laplacian) on five vertices:

sage: def DinverseA(g):
....:   for i in range(g.order()):
....:       A.rescale_row(i, 1/len(A.nonzero_positions_in_row(i)))
....:   return A
sage: g=graphs.cospectral_graphs(5, matrix_function=DinverseA, graphs=lambda g: min(g.degree())>0)
sage: sorted(sorted(g.graph6_string() for g in glist) for glist in g)
[['Dlg', 'Ds_']]
sage: g.laplacian_matrix(normalized=True).charpoly()==g.laplacian_matrix(normalized=True).charpoly()
True

fullerenes(order, ipr=False)

Returns a generator which creates fullerene graphs using the buckygen generator (see [buckygen]).

INPUT:

• order - a positive even integer smaller than or equal to 254. This specifies the number of vertices in the generated fullerenes.
• ipr - default: False - if True only fullerenes that satisfy the Isolated Pentagon Rule are generated. This means that no pentagonal faces share an edge.

OUTPUT:

A generator which will produce the fullerene graphs as Sage graphs with an embedding set. These will be simple graphs: no loops, no multiple edges, no directed edges.

EXAMPLES:

There are 1812 isomers of $$\textrm{C}_{60}$$, i.e., 1812 fullerene graphs on 60 vertices:

sage: gen = graphs.fullerenes(60)  # optional buckygen
sage: len(list(gen))  # optional buckygen
1812


However, there is only one IPR fullerene graph on 60 vertices: the famous Buckminster Fullerene:

sage: gen = graphs.fullerenes(60, ipr=True)  # optional buckygen
sage: next(gen)  # optional buckygen
Graph on 60 vertices
sage: next(gen)  # optional buckygen
Traceback (most recent call last):
...
StopIteration


The unique fullerene graph on 20 vertices is isomorphic to the dodecahedron graph.

sage: gen = graphs.fullerenes(20)  # optional buckygen
sage: g = next(gen)  # optional buckygen
sage: g.is_isomorphic(graphs.DodecahedralGraph()) # optional buckygen
True
sage: g.get_embedding()  # optional buckygen
{1: [2, 3, 4],
2: [1, 5, 6],
3: [1, 7, 8],
4: [1, 9, 10],
5: [2, 10, 11],
6: [2, 12, 7],
7: [3, 6, 13],
8: [3, 14, 9],
9: [4, 8, 15],
10: [4, 16, 5],
11: [5, 17, 12],
12: [6, 11, 18],
13: [7, 18, 14],
14: [8, 13, 19],
15: [9, 19, 16],
16: [10, 15, 17],
17: [11, 16, 20],
18: [12, 20, 13],
19: [14, 20, 15],
20: [17, 19, 18]}
sage: g.plot3d(layout='spring')  # optional buckygen
Graphics3d Object


REFERENCE:

 [buckygen] G. Brinkmann, J. Goedgebeur and B.D. McKay, Generation of Fullerenes, Journal of Chemical Information and Modeling, 52(11):2910-2918, 2012.
fusenes(hexagon_count, benzenoids=False)

Returns a generator which creates fusenes and benzenoids using the benzene generator (see [benzene]). Fusenes are planar polycyclic hydrocarbons with all bounded faces hexagons. Benzenoids are fusenes that are subgraphs of the hexagonal lattice.

INPUT:

• hexagon_count - a positive integer smaller than or equal to 30. This specifies the number of hexagons in the generated benzenoids.
• benzenoids - default: False - if True only benzenoids are generated.

OUTPUT:

A generator which will produce the fusenes as Sage graphs with an embedding set. These will be simple graphs: no loops, no multiple edges, no directed edges.

EXAMPLES:

There is a unique fusene with 2 hexagons:

sage: gen = graphs.fusenes(2)  # optional benzene
sage: len(list(gen))  # optional benzene
1


This fusene is naphtalene ($$\textrm{C}_{10}\textrm{H}_{8}$$). In the fusene graph the H-atoms are not stored, so this is a graph on just 10 vertices:

sage: gen = graphs.fusenes(2)  # optional benzene
sage: next(gen)  # optional benzene
Graph on 10 vertices
sage: next(gen)  # optional benzene
Traceback (most recent call last):
...
StopIteration


There are 6505 benzenoids with 9 hexagons:

sage: gen = graphs.fusenes(9, benzenoids=True)  # optional benzene
sage: len(list(gen))  # optional benzene
6505


REFERENCE:

 [benzene] G. Brinkmann, G. Caporossi and P. Hansen, A Constructive Enumeration of Fusenes and Benzenoids, Journal of Algorithms, 45:155-166, 2002.
static line_graph_forbidden_subgraphs()

Returns the 9 forbidden subgraphs of a line graph.

The graphs are returned in the ordering given by the Wikipedia drawing, read from left to right and from top to bottom.

EXAMPLES:

sage: graphs.line_graph_forbidden_subgraphs()
[Claw graph: Graph on 4 vertices,
Graph on 6 vertices,
Graph on 6 vertices,
Graph on 5 vertices,
Graph on 6 vertices,
Graph on 6 vertices,
Graph on 6 vertices,
Graph on 6 vertices,
Graph on 5 vertices]

nauty_geng(options='', debug=False)

Returns a generator which creates graphs from nauty’s geng program.

INPUT:

• options - a string passed to geng as if it was run at a system command line. At a minimum, you must pass the number of vertices you desire. Sage expects the graphs to be in nauty’s “graph6” format, do not set an option to change this default or results will be unpredictable.
• debug - default: False - if True the first line of geng’s output to standard error is captured and the first call to the generator’s next() function will return this line as a string. A line leading with “>A” indicates a successful initiation of the program with some information on the arguments, while a line beginning with “>E” indicates an error with the input.

The possible options, obtained as output of geng --help:

     n    : the number of vertices
mine:maxe : a range for the number of edges
#:0 means '# or more' except in the case 0:0
res/mod : only generate subset res out of subsets 0..mod-1

-c    : only write connected graphs
-C    : only write biconnected graphs
-t    : only generate triangle-free graphs
-f    : only generate 4-cycle-free graphs
-b    : only generate bipartite graphs
(-t, -f and -b can be used in any combination)
-m    : save memory at the expense of time (only makes a
difference in the absence of -b, -t, -f and n <= 28).
-d#   : a lower bound for the minimum degree
-D#   : a upper bound for the maximum degree
-v    : display counts by number of edges
-l    : canonically label output graphs

-q    : suppress auxiliary output (except from -v)


Options which cause geng to use an output format different than the graph6 format are not listed above (-u, -g, -s, -y, -h) as they will confuse the creation of a Sage graph. The res/mod option can be useful when using the output in a routine run several times in parallel.

OUTPUT:

A generator which will produce the graphs as Sage graphs. These will be simple graphs: no loops, no multiple edges, no directed edges.

Graph.is_strongly_regular() – tests whether a graph is strongly regular and/or returns its parameters.

EXAMPLES:

The generator can be used to construct graphs for testing, one at a time (usually inside a loop). Or it can be used to create an entire list all at once if there is sufficient memory to contain it.

sage: gen = graphs.nauty_geng("2")
sage: next(gen)
Graph on 2 vertices
sage: next(gen)
Graph on 2 vertices
sage: next(gen)
Traceback (most recent call last):
...
StopIteration


A list of all graphs on 7 vertices. This agrees with OEIS sequence A000088.

sage: gen = graphs.nauty_geng("7")
sage: len(list(gen))
1044


A list of just the connected graphs on 7 vertices. This agrees with OEIS sequence A001349.

sage: gen = graphs.nauty_geng("7 -c")
sage: len(list(gen))
853


The debug switch can be used to examine geng’s reaction to the input in the options string. We illustrate success. (A failure will be a string beginning with “>E”.) Passing the “-q” switch to geng will supress the indicator of a successful initiation.

sage: gen = graphs.nauty_geng("4", debug=True)
sage: print(next(gen))
>A geng -d0D3 n=4 e=0-6

static petersen_family(generate=False)

Returns the Petersen family

The Petersen family is a collection of 7 graphs which are the forbidden minors of the linklessly embeddable graphs. For more information see the Wikipedia article Petersen_family.

INPUT:

• generate (boolean) – whether to generate the family from the $$\Delta-Y$$ transformations. When set to False (default) a hardcoded version of the graphs (with a prettier layout) is returned.

EXAMPLES:

sage: graphs.petersen_family()
[Petersen graph: Graph on 10 vertices,
Complete graph: Graph on 6 vertices,
Multipartite Graph with set sizes [3, 3, 1]: Graph on 7 vertices,
Graph on 8 vertices,
Graph on 9 vertices,
Graph on 7 vertices,
Graph on 8 vertices]


The two different inputs generate the same graphs:

sage: F1 = graphs.petersen_family(generate=False)
sage: F2 = graphs.petersen_family(generate=True)
sage: F1 = [g.canonical_label().graph6_string() for g in F1]
sage: F2 = [g.canonical_label().graph6_string() for g in F2]
sage: set(F1) == set(F2)
True

planar_graphs(order, minimum_degree=None, minimum_connectivity=None, exact_connectivity=False, only_bipartite=False, dual=False)

An iterator over connected planar graphs using the plantri generator.

This uses the plantri generator (see [plantri]) which is available through the optional package plantri.

Note

The non-3-connected graphs will be returned several times, with all its possible embeddings.

INPUT:

• order - a positive integer smaller than or equal to 64. This specifies the number of vertices in the generated graphs.
• minimum_degree - default: None - a value $$\geq 1$$ and $$\leq 5$$, or None. This specifies the minimum degree of the generated graphs. If this is None and the order is 1, then this is set to 0. If this is None and the minimum connectivity is specified, then this is set to the same value as the minimum connectivity. If the minimum connectivity is also equal to None, then this is set to 1.
• minimum_connectivity - default: None - a value $$\geq 1$$ and $$\leq 3$$, or None. This specifies the minimum connectivity of the generated graphs. If this is None and the minimum degree is specified, then this is set to the minimum of the minimum degree and 3. If the minimum degree is also equal to None, then this is set to 1.
• exact_connectivity - default: False - if True only graphs with exactly the specified connectivity will be generated. This option cannot be used with minimum_connectivity=3, or if the minimum connectivity is not explicitely set.
• only_bipartite - default: False - if True only bipartite graphs will be generated. This option cannot be used for graphs with a minimum degree larger than 3.
• dual - default: False - if True return instead the planar duals of the generated graphs.

OUTPUT:

An iterator which will produce all planar graphs with the given number of vertices as Sage graphs with an embedding set. These will be simple graphs (no loops, no multiple edges, no directed edges) unless the option dual=True is used.

EXAMPLES:

There are 6 planar graphs on 4 vertices:

sage: gen = graphs.planar_graphs(4)  # optional plantri
sage: len(list(gen))  # optional plantri
6


Three of these planar graphs are bipartite:

sage: gen = graphs.planar_graphs(4, only_bipartite=True)  # optional plantri
sage: len(list(gen))  # optional plantri
3


Setting dual=True gives the planar dual graphs:

sage: gen = graphs.planar_graphs(4, dual=True)  # optional plantri
sage: [u for u in list(gen)]  # optional plantri
[Graph on 4 vertices,
Multi-graph on 3 vertices,
Multi-graph on 2 vertices,
Looped multi-graph on 2 vertices,
Looped multi-graph on 1 vertex,
Looped multi-graph on 1 vertex]


The cycle of length 4 is the only 2-connected bipartite planar graph on 4 vertices:

sage: l = list(graphs.planar_graphs(4, minimum_connectivity=2, only_bipartite=True))  # optional plantri
sage: l.get_embedding()  # optional plantri
{1: [2, 3],
2: [1, 4],
3: [1, 4],
4: [2, 3]}


There is one planar graph with one vertex. This graph obviously has minimum degree equal to 0:

sage: list(graphs.planar_graphs(1))  # optional plantri
[Graph on 1 vertex]
sage: list(graphs.planar_graphs(1, minimum_degree=1))  # optional plantri
[]


REFERENCE:

 [plantri] (1, 2, 3) G. Brinkmann and B.D. McKay, Fast generation of planar graphs, MATCH-Communications in Mathematical and in Computer Chemistry, 58(2):323-357, 2007.
quadrangulations(order, minimum_degree=None, minimum_connectivity=None, no_nonfacial_quadrangles=False, dual=False)

An iterator over planar quadrangulations using the plantri generator.

This uses the plantri generator (see [plantri]) which is available through the optional package plantri.

INPUT:

• order - a positive integer smaller than or equal to 64. This specifies the number of vertices in the generated quadrangulations.
• minimum_degree - default: None - a value $$\geq 2$$ and $$\leq 3$$, or None. This specifies the minimum degree of the generated quadrangulations. If this is None and the minimum connectivity is specified, then this is set to the same value as the minimum connectivity. If the minimum connectivity is also equal to None, then this is set to 2.
• minimum_connectivity - default: None - a value $$\geq 2$$ and $$\leq 3$$, or None. This specifies the minimum connectivity of the generated quadrangulations. If this is None and the option no_nonfacial_quadrangles is set to True, then this is set to 3. Otherwise if this is None and the minimum degree is specified, then this is set to the minimum degree. If the minimum degree is also equal to None, then this is set to 3.
• no_nonfacial_quadrangles - default: False - if True only quadrangulations with no non-facial quadrangles are generated. This option cannot be used if minimum_connectivity is set to 2.
• dual - default: False - if True return instead the planar duals of the generated graphs.

OUTPUT:

An iterator which will produce all planar quadrangulations with the given number of vertices as Sage graphs with an embedding set. These will be simple graphs (no loops, no multiple edges, no directed edges).

EXAMPLES:

The cube is the only 3-connected planar quadrangulation on 8 vertices:

sage: gen = graphs.quadrangulations(8, minimum_connectivity=3)  # optional plantri
sage: g = next(gen)                                            # optional plantri
sage: g.is_isomorphic(graphs.CubeGraph(3))                      # optional plantri
True
sage: next(gen)                                                # optional plantri
Traceback (most recent call last):
...
StopIteration


An overview of the number of quadrangulations on up to 12 vertices. This agrees with OEIS sequence A113201:

sage: for i in range(4,13):                          # optional plantri
....:     L =  len(list(graphs.quadrangulations(i))) # optional plantri
....:     print("{:2d}   {:3d}".format(i,L))         # optional plantri
4     1
5     1
6     2
7     3
8     9
9    18
10    62
11   198
12   803


There are 2 planar quadrangulation on 12 vertices that do not have a non-facial quadrangle:

sage: len([g for g in graphs.quadrangulations(12, no_nonfacial_quadrangles=True)])  # optional plantri
2


Setting dual=True gives the planar dual graphs:

sage: [len(g) for g in graphs.quadrangulations(12, no_nonfacial_quadrangles=True, dual=True)]  # optional plantri
[10, 10]

static strongly_regular_graph(v, k, l, mu=-1, existence=False, check=True)

Return a $$(v,k,\lambda,\mu)$$-strongly regular graph.

This function relies partly on Andries Brouwer’s database of strongly regular graphs. See the documentation of sage.graphs.strongly_regular_db for more information.

INPUT:

• v,k,l,mu (integers) – note that mu, if unspecified, is automatically determined from v,k,l.

• existence (boolean;False) – instead of building the graph, return:

• True – meaning that a $$(v,k,\lambda,\mu)$$-strongly regular graph exists.
• Unknown – meaning that Sage does not know if such a strongly regular graph exists (see sage.misc.unknown).
• False – meaning that no such strongly regular graph exists.
• check – (boolean) Whether to check that output is correct before returning it. As this is expected to be useless (but we are cautious guys), you may want to disable it whenever you want speed. Set to True by default.

EXAMPLES:

Petersen’s graph from its set of parameters:

sage: graphs.strongly_regular_graph(10,3,0,1,existence=True)
True
sage: graphs.strongly_regular_graph(10,3,0,1)
complement(Johnson graph with parameters 5,2): Graph on 10 vertices


Now without specifying $$\mu$$:

sage: graphs.strongly_regular_graph(10,3,0)
complement(Johnson graph with parameters 5,2): Graph on 10 vertices


An obviously infeasible set of parameters:

sage: graphs.strongly_regular_graph(5,5,5,5,existence=True)
False
sage: graphs.strongly_regular_graph(5,5,5,5)
Traceback (most recent call last):
...
ValueError: There exists no (5, 5, 5, 5)-strongly regular graph


An set of parameters proved in a paper to be infeasible:

sage: graphs.strongly_regular_graph(324,57,0,12,existence=True)
False
sage: graphs.strongly_regular_graph(324,57,0,12)
Traceback (most recent call last):
...
EmptySetError: Andries Brouwer's database reports that no (324, 57, 0,
12)-strongly regular graph exists. Comments: <a
href="srgtabrefs.html#GavrilyukMakhnev05">Gavrilyuk & Makhnev</a> ...


A set of parameters unknown to be realizable in Andries Brouwer’s database:

sage: graphs.strongly_regular_graph(324,95,22,30,existence=True)
Unknown
sage: graphs.strongly_regular_graph(324,95,22,30)
Traceback (most recent call last):
...
RuntimeError: Andries Brouwer's database reports that no
(324, 95, 22, 30)-strongly regular graph is known to exist.


A large unknown set of parameters (not in Andries Brouwer’s database):

sage: graphs.strongly_regular_graph(1394,175,0,25,existence=True)
Unknown
sage: graphs.strongly_regular_graph(1394,175,0,25)
Traceback (most recent call last):
...
RuntimeError: Sage cannot figure out if a (1394, 175, 0, 25)-strongly
regular graph exists.


Test the Claw bound (see 3.D of [BvL84]):

sage: graphs.strongly_regular_graph(2058,242,91,20,existence=True)
False

static trees(vertices)

Returns a generator of the distinct trees on a fixed number of vertices.

INPUT:

• vertices - the size of the trees created.

OUTPUT:

A generator which creates an exhaustive, duplicate-free listing of the connected free (unlabeled) trees with vertices number of vertices. A tree is a graph with no cycles.

ALGORITHM:

Uses an algorithm that generates each new tree in constant time. See the documentation for, and implementation of, the sage.graphs.trees module, including a citation.

EXAMPLES:

We create an iterator, then loop over its elements.

sage: tree_iterator = graphs.trees(7)
sage: for T in tree_iterator:
....:     print(T.degree_sequence())
[2, 2, 2, 2, 2, 1, 1]
[3, 2, 2, 2, 1, 1, 1]
[3, 2, 2, 2, 1, 1, 1]
[4, 2, 2, 1, 1, 1, 1]
[3, 3, 2, 1, 1, 1, 1]
[3, 3, 2, 1, 1, 1, 1]
[4, 3, 1, 1, 1, 1, 1]
[3, 2, 2, 2, 1, 1, 1]
[4, 2, 2, 1, 1, 1, 1]
[5, 2, 1, 1, 1, 1, 1]
[6, 1, 1, 1, 1, 1, 1]


The number of trees on the first few vertex counts. This is sequence A000055 in Sloane’s OEIS.

sage: [len(list(graphs.trees(i))) for i in range(0, 15)]
[1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159]

triangulations(order, minimum_degree=None, minimum_connectivity=None, exact_connectivity=False, only_eulerian=False, dual=False)

An iterator over connected planar triangulations using the plantri generator.

This uses the plantri generator (see [plantri]) which is available through the optional package plantri.

INPUT:

• order - a positive integer smaller than or equal to 64. This specifies the number of vertices in the generated triangulations.
• minimum_degree - default: None - a value $$\geq 3$$ and $$\leq 5$$, or None. This specifies the minimum degree of the generated triangulations. If this is None and the minimum connectivity is specified, then this is set to the same value as the minimum connectivity. If the minimum connectivity is also equal to None, then this is set to 3.
• minimum_connectivity - default: None - a value $$\geq 3$$ and $$\leq 5$$, or None. This specifies the minimum connectivity of the generated triangulations. If this is None and the minimum degree is specified, then this is set to the minimum of the minimum degree and 3. If the minimum degree is also equal to None, then this is set to 3.
• exact_connectivity - default: False - if True only triangulations with exactly the specified connectivity will be generated. This option cannot be used with minimum_connectivity=3, or if the minimum connectivity is not explicitely set.
• only_eulerian - default: False - if True only Eulerian triangulations will be generated. This option cannot be used if the minimum degree is explicitely set to anything else than 4.
• dual - default: False - if True return instead the planar duals of the generated graphs.

OUTPUT:

An iterator which will produce all planar triangulations with the given number of vertices as Sage graphs with an embedding set. These will be simple graphs (no loops, no multiple edges, no directed edges).

EXAMPLES:

The unique planar embedding of the $$K_4$$ is the only planar triangulations on 4 vertices:

sage: gen = graphs.triangulations(4)    # optional plantri
sage: [g.get_embedding() for g in gen]  # optional plantri
[{1: [2, 3, 4], 2: [1, 4, 3], 3: [1, 2, 4], 4: [1, 3, 2]}]


but, of course, this graph is not Eulerian:

sage: gen = graphs.triangulations(4, only_eulerian=True)  # optional plantri
sage: len(list(gen))                                      # optional plantri
0


The unique Eulerian triangulation on 6 vertices is isomorphic to the octahedral graph.

sage: gen = graphs.triangulations(6, only_eulerian=True)  # optional plantri
sage: g = next(gen)                                       # optional plantri
sage: g.is_isomorphic(graphs.OctahedralGraph())           # optional plantri
True


An overview of the number of 5-connected triangulations on up to 22 vertices. This agrees with OEIS sequence A081621:

sage: for i in range(12, 23):                                             # optional plantri
....:     L = len(list(graphs.triangulations(i, minimum_connectivity=5))) # optional plantri
....:     print("{}   {:3d}".format(i,L))                                 # optional plantri
12     1
13     0
14     1
15     1
16     3
17     4
18    12
19    23
20    71
21   187
22   627


The minimum connectivity can be at most the minimum degree:

sage: gen = next(graphs.triangulations(10, minimum_degree=3, minimum_connectivity=5))  # optional plantri
Traceback (most recent call last):
...
ValueError: Minimum connectivity can be at most the minimum degree.


There are 5 triangulations with 9 vertices and minimum degree equal to 4 that are 3-connected, but only one of them is not 4-connected:

sage: len([g for g in graphs.triangulations(9, minimum_degree=4, minimum_connectivity=3)]) # optional plantri
5
sage: len([g for g in graphs.triangulations(9, minimum_degree=4, minimum_connectivity=3, exact_connectivity=True)]) # optional plantri
1


Setting dual=True gives the planar dual graphs:

sage: [len(g) for g in graphs.triangulations(9, minimum_degree=4, minimum_connectivity=3, dual=True)]  # optional plantri
[14, 14, 14, 14, 14]

sage.graphs.graph_generators.canaug_traverse_edge(g, aut_gens, property, dig=False, loops=False, implementation='c_graph', sparse=True)

Main function for exhaustive generation. Recursive traversal of a canonically generated tree of isomorph free graphs satisfying a given property.

INPUT:

• g - current position on the tree.
• aut_gens - list of generators of Aut(g), in list notation.
• property - check before traversing below g.

EXAMPLES:

sage: from sage.graphs.graph_generators import canaug_traverse_edge
sage: G = Graph(3)
sage: list(canaug_traverse_edge(G, [], lambda x: True))
[Graph on 3 vertices, ... Graph on 3 vertices]


The best way to access this function is through the graphs() iterator:

Print graphs on 3 or less vertices.

sage: for G in graphs(3):
....:     print(G)
Graph on 3 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 3 vertices


Print digraphs on 3 or less vertices.

sage: for G in digraphs(3):
....:     print(G)
Digraph on 3 vertices
Digraph on 3 vertices
...
Digraph on 3 vertices
Digraph on 3 vertices

sage.graphs.graph_generators.canaug_traverse_vert(g, aut_gens, max_verts, property, dig=False, loops=False, implementation='c_graph', sparse=True)

Main function for exhaustive generation. Recursive traversal of a canonically generated tree of isomorph free (di)graphs satisfying a given property.

INPUT:

• g - current position on the tree.
• aut_gens - list of generators of Aut(g), in list notation.
• max_verts - when to retreat.
• property - check before traversing below g.
• degree_sequence - specify a degree sequence to try to obtain.

EXAMPLES:

sage: from sage.graphs.graph_generators import canaug_traverse_vert
sage: list(canaug_traverse_vert(Graph(), [], 3, lambda x: True))
[Graph on 0 vertices, ... Graph on 3 vertices]


The best way to access this function is through the graphs() iterator:

Print graphs on 3 or less vertices.

sage: for G in graphs(3, augment='vertices'):
....:    print(G)
Graph on 0 vertices
Graph on 1 vertex
Graph on 2 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 3 vertices
Graph on 2 vertices
Graph on 3 vertices


Print digraphs on 2 or less vertices.

sage: for D in digraphs(2, augment='vertices'):
....:     print(D)
Digraph on 0 vertices
Digraph on 1 vertex
Digraph on 2 vertices
Digraph on 2 vertices
Digraph on 2 vertices

sage.graphs.graph_generators.check_aut(aut_gens, cut_vert, n)

Helper function for exhaustive generation.

At the start, check_aut is given a set of generators for the automorphism group, aut_gens. We already know we are looking for an element of the auto- morphism group that sends cut_vert to n, and check_aut generates these for the canaug_traverse function.

EXAMPLES:

Note that the last two entries indicate that none of the automorphism group has yet been searched - we are starting at the identity [0, 1, 2, 3] and so far that is all we have seen. We return automorphisms mapping 2 to 3:

sage: from sage.graphs.graph_generators import check_aut
sage: list( check_aut( [ [0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3] ], 2, 3))
[[1, 0, 3, 2], [1, 2, 3, 0]]

sage.graphs.graph_generators.check_aut_edge(aut_gens, cut_edge, i, j, n, dig=False)

Helper function for exhaustive generation.

At the start, check_aut_edge is given a set of generators for the automorphism group, aut_gens. We already know we are looking for an element of the auto- morphism group that sends cut_edge to {i, j}, and check_aut generates these for the canaug_traverse function.

EXAMPLES:

Note that the last two entries indicate that none of the automorphism group has yet been searched - we are starting at the identity [0, 1, 2, 3] and so far that is all we have seen. We return automorphisms mapping 2 to 3:

sage: from sage.graphs.graph_generators import check_aut
sage: list( check_aut( [ [0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3] ], 2, 3))
[[1, 0, 3, 2], [1, 2, 3, 0]]
`