Various small graphs

The methods defined here appear in sage.graphs.graph_generators.

sage.graphs.generators.smallgraphs.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 – integer (default: 1); 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
sage.graphs.generators.smallgraphs.Balaban11Cage(embedding=1)

Return the Balaban 11-cage.

For more information, see the Wikipedia article Balaban_11-cage.

INPUT:

  • embedding – integer (default: 1); 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
sage.graphs.generators.smallgraphs.BidiakisCube()

Return the Bidiakis cube.

For more information, see the Wikipedia article 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
sage.graphs.generators.smallgraphs.BiggsSmithGraph(embedding=1)

Return the Biggs-Smith graph.

For more information, see the Wikipedia article Biggs-Smith_graph.

INPUT:

  • embedding – integer (default: 1); 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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.BrinkmannGraph()

Return the Brinkmann graph.

For more information, see the Wikipedia article 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
sage: G.radius()
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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.BuckyBall()

Return 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()  # long time
sage: g.plot(vertex_labels=False, vertex_size=10).show()  # long time
sage.graphs.generators.smallgraphs.CameronGraph()

Return the Cameron graph.

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

For more information on the Cameron graph, see https://www.win.tue.nl/~aeb/graphs/Cameron.html.

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)
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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 – integer (default: 1); 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
sage.graphs.generators.smallgraphs.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
sage: G.radius(); G.diameter(); G.girth()
2
2
4
sage.graphs.generators.smallgraphs.ClebschGraph()

Return the Clebsch graph.

See the Wikipedia article Clebsch_graph for more information.

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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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()
sage.graphs.generators.smallgraphs.DurerGraph()

Return the Dürer graph.

For more information, see the Wikipedia article D%C3%BCrer_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
sage.graphs.generators.smallgraphs.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).

sage.graphs.generators.smallgraphs.EllinghamHorton54Graph()

Return the Ellingham-Horton 54-graph.

For more information, see the Wikipedia article Ellingham-Horton_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
sage.graphs.generators.smallgraphs.EllinghamHorton78Graph()

Return the Ellingham-Horton 78-graph.

For more information, see the Wikipedia article Ellingham%E2%80%93Horton_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
sage.graphs.generators.smallgraphs.ErreraGraph()

Return the Errera graph.

For more information, see the Wikipedia article 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
sage: G.radius()
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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.FranklinGraph()

Return the Franklin graph.

For more information, see the Wikipedia article 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
sage: G.radius()
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
sage.graphs.generators.smallgraphs.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 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
sage.graphs.generators.smallgraphs.GoldnerHararyGraph()

Return the Goldner-Harary graph.

For more information, see the Wikipedia article Goldner%E2%80%93Harary_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
sage: G.radius()
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
sage.graphs.generators.smallgraphs.GolombGraph()

Return the Golomb graph.

See the Wikipedia article Golomb_graph for more information.

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[0]-v[0])**2 + (u[1]-v[1])**2
sage: all(dist2(pos[u], pos[v]) == 1 for u, v in G.edge_iterator(labels=None))
True
sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.GrayGraph(embedding=1)

Return the Gray graph.

See the Wikipedia article Gray_graph.

INPUT:

  • embedding – integer (default: 1); 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
sage.graphs.generators.smallgraphs.GritsenkoGraph()

Return SRG(65, 32, 15, 16) constructed by Gritsenko

We took the adjacency matrix from O.Gritsenko’s [Gri2021] and extracted orbits of the automorphism group on the edges.

EXAMPLES:

sage: H = graphs.GritsenkoGraph(); H
Gritsenko strongly regular graph: Graph on 65 vertices
sage: H.is_strongly_regular(parameters=True)
(65, 32, 15, 16)
sage.graphs.generators.smallgraphs.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
sage: G.radius()
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
sage.graphs.generators.smallgraphs.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 (default: True); 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.

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))
....:     nv.discard(0)
....:     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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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 – integer (default: 1); 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
sage.graphs.generators.smallgraphs.HarriesWongGraph(embedding=1)

Return the Harries-Wong Graph.

See the Wikipedia article Harries-Wong_graph.

About the default embedding:

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 – integer (default: 1); 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()  # long time
sage.graphs.generators.smallgraphs.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, MobiusKantorGraph()). It is nonplanar and Hamiltonian. It has diameter 3, radius 3, girth 6, and 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
sage.graphs.generators.smallgraphs.HerschelGraph()

Return the Herschel graph.

For more information, see the Wikipedia article 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
sage: G.radius()
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
sage.graphs.generators.smallgraphs.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].

See also the Wikipedia article Higman–Sims_graph.

INPUT:

  • relabel – boolean (default: True); whether to relabel the vertices 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()[1]
sage: K.is_simple()
True
sage: g//K.order()
2

AUTHOR:

  • Rob Beezer (2009-10-24)

sage.graphs.generators.smallgraphs.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
sage: g.radius()
3
sage: g.diameter()
4
sage: g.automorphism_group().cardinality()
48
sage.graphs.generators.smallgraphs.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()[1]  # random
(-0.844..., 0.535...)
sage: HS = graphs.HoffmanSingletonGraph()
sage: HS.layout()[1]  # random
(-0.904..., 0.425...)
sage.graphs.generators.smallgraphs.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
sage: g.radius()
3
sage: g.diameter()
3
sage: g.girth()
5
sage: g.automorphism_group().cardinality()
54
sage.graphs.generators.smallgraphs.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
sage: g.radius()
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
sage.graphs.generators.smallgraphs.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

sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.KittellGraph()

Return the Kittell Graph.

For more information, see the Wolfram page about the Kittel Graph.

EXAMPLES:

sage: g = graphs.KittellGraph()
sage: g.order()
23
sage: g.size()
63
sage: g.radius()
3
sage: g.diameter()
4
sage: g.girth()
3
sage: g.chromatic_number()
4
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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, betweenness 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
sage.graphs.generators.smallgraphs.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 - internet
sage: g.order()  # optional - internet
266
sage: g.size()  # optional - internet
1463
sage: g.girth()  # optional - internet
5
sage: g.is_vertex_transitive()  # optional - internet
True
sage: g.is_distance_regular()  # optional - internet
True
sage.graphs.generators.smallgraphs.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 – integer (default: 1); 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
sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.McGeeGraph(embedding=2)

Return the McGee Graph.

See the Wikipedia article McGee_graph.

INPUT:

  • embedding – integer (default: 2); 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()  # long time
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage: g.radius()
7
sage: g.diameter()
8
sage: g.girth()
4
sage: g.chromatic_number()
3
sage: g.is_hamiltonian()  # long time
False
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.MoserSpindle()

Return the Moser spindle.

For more information, see the Wikipedia article 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
sage: G.radius()
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
sage.graphs.generators.smallgraphs.NauruGraph(embedding=2)

Return the Nauru Graph.

See the Wikipedia article Nauru_graph.

INPUT:

  • embedding – integer (default: 2); 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()  # long time
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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])
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.RobertsonGraph()

Return the Robertson graph.

See the Wikipedia article 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
sage.graphs.generators.smallgraphs.SchlaefliGraph()

Return the Schläfli graph.

The Schläfli graph is the only strongly regular graphs of parameters \((27,16,10,8)\) (see [GR2001]).

For more information, see the Wikipedia article Schläfli_graph.

See also

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
sage.graphs.generators.smallgraphs.ShrikhandeGraph()

Return the Shrikhande graph.

For more information, see the MathWorld article on the Shrikhande graph or the Wikipedia article Shrikhande_graph.

See also

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).

sage.graphs.generators.smallgraphs.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.

See also

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)
sage.graphs.generators.smallgraphs.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
sage: g.radius()
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
sage.graphs.generators.smallgraphs.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 [Hub1975].

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)
sage.graphs.generators.smallgraphs.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.

For more information on the Sylvester graph, see https://www.win.tue.nl/~aeb/graphs/Sylvester.html.

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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.TietzeGraph()

Return the Tietze Graph.

For more information on the Tietze Graph, see the Wikipedia article Tietze%27s_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
sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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()
sage.graphs.generators.smallgraphs.TutteCoxeterGraph(embedding=2)

Return the Tutte-Coxeter graph.

See the Wikipedia article Tutte-Coxeter_graph.

INPUT:

  • embedding – integer (default: 2); 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()  # long time
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.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)
sage.graphs.generators.smallgraphs.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 [FK1991].

EXAMPLES:

sage: G=graphs.U42Graph540()  # optional - gap_packages (grape)
sage: G.is_strongly_regular(parameters=True)  # optional - gap_packages (grape)
(540, 187, 58, 68)
sage.graphs.generators.smallgraphs.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()
sage.graphs.generators.smallgraphs.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
sage.graphs.generators.smallgraphs.WellsGraph()

Return the Wells graph.

For more information on the Wells graph (also called Armanios-Wells graph), see this page.

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
sage.graphs.generators.smallgraphs.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