Common digraphs

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

sage: g = digraphs.Circuit(15)

To get a circulant graph on 10 vertices in which a vertex \(i\) has \(i+2\) and \(i+3\) as outneighbors:

sage: p = digraphs.Circulant(10,[2,3])

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

ButterflyGraph()	Return a \(n\)-dimensional butterfly graph.
Circuit()	Return the circuit on \(n\) vertices.
Circulant()	Return a circulant digraph on \(n\) vertices from a set of integers.
Complete()	Return a complete digraph on \(n\) vertices.
DeBruijn()	Return the De Bruijn digraph with parameters \(k,n\).
GeneralizedDeBruijn()	Return the generalized de Bruijn digraph of order \(n\) and degree \(d\).
ImaseItoh()	Return the digraph of Imase and Itoh of order \(n\) and degree \(d\).
Kautz()	Return the Kautz digraph of degree \(d\) and diameter \(D\).
nauty_directg()	Return an iterator yielding digraphs using nauty’s directg program.
Paley()	Return a Paley digraph on \(q\) vertices.
Path()	Return a directed path on \(n\) vertices.
RandomDirectedAcyclicGraph()	Return a random (weighted) directed acyclic graph of order \(n\).
RandomDirectedGNC()	Return a random growing network with copying (GNC) digraph with \(n\) vertices.
RandomDirectedGNM()	Return a random labelled digraph on \(n\) nodes and \(m\) arcs.
RandomDirectedGNP()	Return a random digraph on \(n\) nodes.
RandomDirectedGN()	Return a random growing network (GN) digraph with \(n\) vertices.
RandomDirectedGNR()	Return a random growing network with redirection (GNR) digraph.
RandomSemiComplete()	Return a random semi-complete digraph of order \(n\).
RandomTournament()	Return a random tournament on \(n\) vertices.
TransitiveTournament()	Return a transitive tournament on \(n\) vertices.
tournaments_nauty()	Iterator over all tournaments on \(n\) vertices using Nauty.

AUTHORS:

• Robert L. Miller (2006)

• Emily A. Kirkman (2006)

• Michael C. Yurko (2009)

• David Coudert (2012)

Functions and methods

class sage.graphs.digraph_generators.DiGraphGenerators

Bases: object

A class consisting of constructors for several common digraphs, including orderly generation of isomorphism class representatives.

A list of all graphs and graph structures in this database is available via tab completion. Type “digraphs.” and then hit tab to see which graphs are available.

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

The constructors currently in this class include:

Random Directed Graphs:
    - RandomDirectedAcyclicGraph
    - RandomDirectedGN
    - RandomDirectedGNC
    - RandomDirectedGNP
    - RandomDirectedGNM
    - RandomDirectedGNR
    - RandomTournament
    - RandomSemiComplete

Families of Graphs:
    - Complete
    - DeBruijn
    - GeneralizedDeBruijn
    - Kautz
    - Path
    - ImaseItoh
    - RandomTournament
    - TransitiveTournament
    - tournaments_nauty

ORDERLY GENERATION: digraphs(vertices, property=lambda x: True, augment=’edges’, size=None)

Accesses the generator of isomorphism class representatives [McK1998]. Iterates over distinct, exhaustive representatives.

INPUT:

• vertices – natural number or None to infinitely generate bigger and bigger digraphs.

• property – any property to be tested on digraphs before generation

• augment – choices:

  • 'vertices' – augments by adding a vertex, and edges incident to that vertex. In this case, all digraphs on up to n=vertices are generated. If for any digraph 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 digraphs with that property. If this does not hold, then all the digraphs generated will satisfy the property, but there will be some missing.

  • 'edges' – augments a fixed number of vertices by adding one edge. In this case, all digraphs 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 digraphs with that property. If this does not hold, then all the digraphs generated will satisfy the property, but there will be some missing.

• implementation – which underlying implementation to use (see DiGraph?)

• sparse – boolean (default: True); whether to use a sparse or dense data structure. See the documentation of Graph.

EXAMPLES:

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

Print digraphs on 3 vertices:

sage: for D in digraphs(3):
....:     print(D)
Digraph on 3 vertices
Digraph on 3 vertices
...
Digraph on 3 vertices
Digraph on 3 vertices

Generate all digraphs with 4 vertices and 3 edges:

sage: L = digraphs(4, size=3)
sage: len(list(L))
13

Generate all digraphs with 4 vertices and up to 3 edges:

sage: L = list(digraphs(4, lambda G: G.size() <= 3))
sage: len(L)
20
sage: graphs_list.show_graphs(L)        # long time                             # needs sage.plot

Generate all digraphs with degree at most 2, up to 5 vertices:

sage: property = lambda G: (max([G.degree(v) for v in G] + [0]) <= 2)
sage: L = list(digraphs(5, property, augment='vertices'))
sage: len(L)
75

Generate digraphs on the fly (see http://oeis.org/classic/A000273):

sage: for i in range(5):
....:     print(len(list(digraphs(i))))
1
1
3
16
218

ButterflyGraph(n, vertices='strings')

Return a \(n\)-dimensional butterfly graph.

The vertices consist of pairs \((v, i)\), where \(v\) is an \(n\)-dimensional tuple (vector) with binary entries (or a string representation of such) and \(i\) is an integer in \([0..n]\). A directed edge goes from \((v, i)\) to \((w, i + 1)\) if \(v\) and \(w\) are identical except for possibly when \(v[i] \neq w[i]\).

A butterfly graph has \((2^n)(n+1)\) vertices and \(n2^{n+1}\) edges.

INPUT:

• n – a non negative integer; the dimension of the butterfly graph

• vertices – string (default: 'strings'); specifies whether the vertices are zero-one strings (default) or tuples over GF(2) (vertices='vectors')

EXAMPLES:

sage: digraphs.ButterflyGraph(2).edges(sort=True, labels=False)
[(('00', 0), ('00', 1)),
 (('00', 0), ('10', 1)),
 (('00', 1), ('00', 2)),
 (('00', 1), ('01', 2)),
 (('01', 0), ('01', 1)),
 (('01', 0), ('11', 1)),
 (('01', 1), ('00', 2)),
 (('01', 1), ('01', 2)),
 (('10', 0), ('00', 1)),
 (('10', 0), ('10', 1)),
 (('10', 1), ('10', 2)),
 (('10', 1), ('11', 2)),
 (('11', 0), ('01', 1)),
 (('11', 0), ('11', 1)),
 (('11', 1), ('10', 2)),
 (('11', 1), ('11', 2))]
sage: digraphs.ButterflyGraph(2, vertices='vectors').edges(sort=True,       # needs sage.modules sage.rings.finite_rings
....:                                                      labels=False)
[(((0, 0), 0), ((0, 0), 1)),
(((0, 0), 0), ((1, 0), 1)),
(((0, 0), 1), ((0, 0), 2)),
(((0, 0), 1), ((0, 1), 2)),
(((0, 1), 0), ((0, 1), 1)),
(((0, 1), 0), ((1, 1), 1)),
(((0, 1), 1), ((0, 0), 2)),
(((0, 1), 1), ((0, 1), 2)),
(((1, 0), 0), ((0, 0), 1)),
(((1, 0), 0), ((1, 0), 1)),
(((1, 0), 1), ((1, 0), 2)),
(((1, 0), 1), ((1, 1), 2)),
(((1, 1), 0), ((0, 1), 1)),
(((1, 1), 0), ((1, 1), 1)),
(((1, 1), 1), ((1, 0), 2)),
(((1, 1), 1), ((1, 1), 2))]
sage: pos = digraphs.ButterflyGraph(2).get_pos()
sage: pos['11', 0]
(0, 0)

Circuit(n)

Return the circuit on \(n\) vertices.

The circuit is an oriented CycleGraph.

EXAMPLES:

A circuit is the smallest strongly connected digraph:

sage: circuit = digraphs.Circuit(15)
sage: len(circuit.strongly_connected_components()) == 1
True

Circulant(n, integers)

Return a circulant digraph on \(n\) vertices from a set of integers.

INPUT:

• n – integer; number of vertices

• integers – iterable container (list, set, etc.) of integers such that there is an edge from \(i\) to \(j\) if and only if (j-i)%n in integers

EXAMPLES:

sage: digraphs.Circulant(13,[3,5,7])
Circulant graph ([3, 5, 7]): Digraph on 13 vertices

Complete(n, loops=False)

Return the complete digraph on \(n\) vertices.

INPUT:

• n – integer; number of vertices

• loops – boolean (default: False); whether to add loops or not, i.e., edges from \(u\) to itself

See also

• RandomSemiComplete()

• RandomTournament()

EXAMPLES:

sage: n = 10
sage: G = digraphs.Complete(n); G
Complete digraph: Digraph on 10 vertices
sage: G.size() == n*(n-1)
True
sage: G = digraphs.Complete(n, loops=True); G
Complete digraph with loops: Looped digraph on 10 vertices
sage: G.size() == n*n
True
sage: digraphs.Complete(-1)
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

DeBruijn(k, n, vertices='strings')

Return the De Bruijn digraph with parameters \(k,n\).

The De Bruijn digraph with parameters \(k,n\) is built upon a set of vertices equal to the set of words of length \(n\) from a dictionary of \(k\) letters.

In this digraph, there is an arc \(w_1w_2\) if \(w_2\) can be obtained from \(w_1\) by removing the leftmost letter and adding a new letter at its right end. For more information, see the Wikipedia article De_Bruijn_graph.

INPUT:

• k – two possibilities for this parameter :

  • An integer equal to the cardinality of the alphabet to use, that is, the degree of the digraph to be produced.

  • An iterable object to be used as the set of letters. The degree of the resulting digraph is the cardinality of the set of letters.

• n – integer; length of words in the De Bruijn digraph when vertices == 'strings', and also the diameter of the digraph.

• vertices – string (default: 'strings'); whether the vertices are words over an alphabet (default) or integers (vertices='string')

EXAMPLES:

de Bruijn digraph of degree 2 and diameter 2:

sage: db = digraphs.DeBruijn(2, 2); db                                      # needs sage.combinat
De Bruijn digraph (k=2, n=2): Looped digraph on 4 vertices
sage: db.order(), db.size()                                                 # needs sage.combinat
(4, 8)
sage: db.diameter()                                                         # needs sage.combinat
2

Building a de Bruijn digraph on a different alphabet:

sage: # needs sage.combinat
sage: g = digraphs.DeBruijn(['a', 'b'], 2)
sage: g.vertices(sort=True)
['aa', 'ab', 'ba', 'bb']
sage: g.is_isomorphic(db)
True
sage: g = digraphs.DeBruijn(['AA', 'BB'], 2)
sage: g.vertices(sort=True)
['AA,AA', 'AA,BB', 'BB,AA', 'BB,BB']
sage: g.is_isomorphic(db)
True

GeneralizedDeBruijn(n, d)

Return the generalized de Bruijn digraph of order \(n\) and degree \(d\).

The generalized de Bruijn digraph was defined in [RPK1980] [RPK1983]. It has vertex set \(V=\{0, 1,..., n-1\}\) and there is an arc from vertex \(u \in V\) to all vertices \(v \in V\) such that \(v \equiv (u*d + a) \mod{n}\) with \(0 \leq a < d\).

When \(n = d^{D}\), the generalized de Bruijn digraph is isomorphic to the de Bruijn digraph of degree \(d\) and diameter \(D\).

INPUT:

• n – integer; number of vertices of the digraph (must be at least one)

• d – integer; degree of the digraph (must be at least one)

See also

• sage.graphs.generic_graph.GenericGraph.is_circulant() – checks whether a (di)graph is circulant, and/or returns all possible sets of parameters.

EXAMPLES:

sage: GB = digraphs.GeneralizedDeBruijn(8, 2)
sage: GB.is_isomorphic(digraphs.DeBruijn(2, 3), certificate=True)           # needs sage.combinat
(True, {0: '000', 1: '001', 2: '010', 3: '011',
        4: '100', 5: '101', 6: '110', 7: '111'})

ImaseItoh(n, d)

Return the Imase-Itoh digraph of order \(n\) and degree \(d\).

The Imase-Itoh digraph was defined in [II1983]. It has vertex set \(V=\{0, 1,..., n-1\}\) and there is an arc from vertex \(u \in V\) to all vertices \(v \in V\) such that \(v \equiv (-u*d-a-1) \mod{n}\) with \(0 \leq a < d\).

When \(n = d^{D}\), the Imase-Itoh digraph is isomorphic to the de Bruijn digraph of degree \(d\) and diameter \(D\). When \(n = d^{D-1}(d+1)\), the Imase-Itoh digraph is isomorphic to the Kautz digraph [Kau1968] of degree \(d\) and diameter \(D\).

INPUT:

• n – integer; number of vertices of the digraph (must be greater than or equal to two)

• d – integer; degree of the digraph (must be greater than or equal to one)

EXAMPLES:

sage: II = digraphs.ImaseItoh(8, 2)
sage: II.is_isomorphic(digraphs.DeBruijn(2, 3), certificate=True)           # needs sage.combinat
(True, {0: '010', 1: '011', 2: '000', 3: '001',
        4: '110', 5: '111', 6: '100', 7: '101'})

sage: II = digraphs.ImaseItoh(12, 2)
sage: b,D = II.is_isomorphic(digraphs.Kautz(2, 3), certificate=True)        # needs sage.combinat
sage: b                                                                     # needs sage.combinat
True
sage: D   # random isomorphism                                              # needs sage.combinat
{0: '202', 1: '201', 2: '210', 3: '212', 4: '121',
 5: '120', 6: '102', 7: '101', 8: '010', 9: '012',
 10: '021', 11: '020'}

Kautz(k, D, vertices='strings')

Return the Kautz digraph of degree \(d\) and diameter \(D\).

The Kautz digraph has been defined in [Kau1968]. The Kautz digraph of degree \(d\) and diameter \(D\) has \(d^{D-1}(d+1)\) vertices. This digraph is built from a set of vertices equal to the set of words of length \(D\) over an alphabet of \(d+1\) letters such that consecutive letters are different. There is an arc from vertex \(u\) to vertex \(v\) if \(v\) can be obtained from \(u\) by removing the leftmost letter and adding a new letter, distinct from the rightmost letter of \(u\), at the right end.

The Kautz digraph of degree \(d\) and diameter \(D\) is isomorphic to the Imase-Itoh digraph [II1983] of degree \(d\) and order \(d^{D-1}(d+1)\).

See the Wikipedia article Kautz_graph for more information.

INPUT:

• k – two possibilities for this parameter. In either case the degree must be at least one:

  • An integer equal to the degree of the digraph to be produced, that is, the cardinality of the alphabet to be used minus one.

  • An iterable object to be used as the set of letters. The degree of the resulting digraph is the cardinality of the set of letters minus one.

• D – integer; diameter of the digraph, and length of a vertex label when vertices == 'strings' (must be at least one)

• vertices – string (default: 'strings'); whether the vertices are words over an alphabet (default) or integers (vertices='strings')

EXAMPLES:

sage: # needs sage.combinat
sage: K = digraphs.Kautz(2, 3)
sage: b, D = K.is_isomorphic(digraphs.ImaseItoh(12, 2), certificate=True)
sage: b
True
sage: D  # random isomorphism
{'010': 8, '012': 9, '020': 11, '021': 10, '101': 7,  '102': 6,
 '120': 5, '121': 4, '201': 1, '202': 0, '210': 2, '212': 3}

sage: K = digraphs.Kautz([1,'a','B'], 2)                                    # needs sage.combinat
sage: K.edges(sort=True)                                                    # needs sage.combinat
[('1B', 'B1', '1'), ('1B', 'Ba', 'a'), ('1a', 'a1', '1'),
 ('1a', 'aB', 'B'), ('B1', '1B', 'B'), ('B1', '1a', 'a'),
 ('Ba', 'a1', '1'), ('Ba', 'aB', 'B'), ('a1', '1B', 'B'),
 ('a1', '1a', 'a'), ('aB', 'B1', '1'), ('aB', 'Ba', 'a')]

sage: K = digraphs.Kautz([1,'aA','BB'], 2)                                  # needs sage.combinat
sage: K.edges(sort=True)                                                    # needs sage.combinat
[('1,BB', 'BB,1', '1'), ('1,BB', 'BB,aA', 'aA'),
 ('1,aA', 'aA,1', '1'), ('1,aA', 'aA,BB', 'BB'),
 ('BB,1', '1,BB', 'BB'), ('BB,1', '1,aA', 'aA'),
 ('BB,aA', 'aA,1', '1'), ('BB,aA', 'aA,BB', 'BB'),
 ('aA,1', '1,BB', 'BB'), ('aA,1', '1,aA', 'aA'),
 ('aA,BB', 'BB,1', '1'), ('aA,BB', 'BB,aA', 'aA')]

Paley(q)

Return a Paley digraph on \(q\) vertices.

Parameter \(q\) must be the power of a prime number and congruent to 3 mod 4.

See also

• Wikipedia article Paley_graph

• PaleyGraph()

EXAMPLES:

A Paley digraph has \(n * (n-1) / 2\) edges, its underlying graph is a clique, and so it is a tournament:

sage: g = digraphs.Paley(7); g                                              # needs sage.rings.finite_rings
Paley digraph with parameter 7: Digraph on 7 vertices
sage: g.size() == g.order() * (g.order() - 1) / 2                           # needs sage.rings.finite_rings
True
sage: g.to_undirected().is_clique()                                         # needs sage.rings.finite_rings
True

A Paley digraph is always self-complementary:

sage: g.complement().is_isomorphic(g)                                       # needs sage.rings.finite_rings
True

Path(n)

Return a directed path on \(n\) vertices.

INPUT:

• n – integer; number of vertices in the path

EXAMPLES:

sage: g = digraphs.Path(5)
sage: g.vertices(sort=True)
[0, 1, 2, 3, 4]
sage: g.size()
4
sage: g.automorphism_group().cardinality()                                  # needs sage.groups
1

RandomDirectedAcyclicGraph(n, p, weight_max=None)

Return a random (weighted) directed acyclic graph of order \(n\).

The method starts with the sink vertex and adds vertices one at a time. A vertex is connected only to previously defined vertices, and the probability of each possible connection is given by the probability \(p\). The weight of an edge is a random integer between 1 and weight_max.

INPUT:

• n – number of nodes of the graph

• p – probability of an edge

• weight_max – (default: None); by default, the returned DAG is unweighted. When weight_max is set to a positive integer, edges are assigned a random integer weight between 1 and weight_max.

EXAMPLES:

sage: D = digraphs.RandomDirectedAcyclicGraph(5, .5); D
RandomDAG(5, 0.500000000000000): Digraph on 5 vertices
sage: D.is_directed_acyclic()
True
sage: D = digraphs.RandomDirectedAcyclicGraph(5, .5, weight_max=3); D
RandomWeightedDAG(5, 0.500000000000000, 3): Digraph on 5 vertices
sage: D.is_directed_acyclic()
True

RandomDirectedGN(n, kernel=None, seed=None)

Return a random growing network (GN) digraph with \(n\) vertices.

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen with a preferential attachment model, i.e. probability is proportional to degree. The default attachment kernel is a linear function of degree. The digraph is always a tree, so in particular it is a directed acyclic graph. See [KR2001b] for more details.

INPUT:

• n – integer; number of vertices

• kernel – the attachment kernel (default: identity function)

• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

sage: # needs networkx
sage: D = digraphs.RandomDirectedGN(25)
sage: D.num_verts()
25
sage: D.num_edges()
24
sage: D.is_connected()
True
sage: D.parent() is DiGraph
True
sage: D.show()                      # long time

RandomDirectedGNC(n, seed=None)

Return a random growing network with copying (GNC) digraph with \(n\) vertices.

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen with a preferential attachment model, i.e. probability is proportional to degree. The new vertex is also linked to all of the previously added vertex’s successors. See [KR2005] for more details.

INPUT:

• n – integer; number of vertices

• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

sage: # needs networkx
sage: D = digraphs.RandomDirectedGNC(25)
sage: D.is_directed_acyclic()
True
sage: D.topological_sort()
[24, 23, ..., 1, 0]
sage: D.show()                      # long time

RandomDirectedGNM(n, m, loops=False)

Return a random labelled digraph on \(n\) nodes and \(m\) arcs.

INPUT:

• n – integer; number of vertices

• m – integer; number of edges

• loops – boolean (default: False); whether to allow loops

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

EXAMPLES:

sage: D = digraphs.RandomDirectedGNM(10, 5)
sage: D.num_verts()
10
sage: D.num_edges()
5

With loops:

sage: D = digraphs.RandomDirectedGNM(10, 100, loops = True)
sage: D.num_verts()
10
sage: D.loops()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None), (4, 4, None), (5, 5, None), (6, 6, None), (7, 7, None), (8, 8, None), (9, 9, None)]

RandomDirectedGNP(n, p, loops=False, seed=None)

Return a random digraph on \(n\) nodes.

Each edge is inserted independently with probability \(p\). See [ER1959] and [Gil1959] for more details.

INPUT:

• n – integer; number of nodes of the digraph

• p – float; probability of an edge

• loops – boolean (default: False); whether the random digraph may have loops

• seed – integer (default: None); seed for random number generator

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

EXAMPLES:

sage: D = digraphs.RandomDirectedGNP(10, .2)
sage: D.num_verts()
10
sage: D.parent() is DiGraph
True

RandomDirectedGNR(n, p, seed=None)

Return a random growing network with redirection (GNR) digraph with \(n\) vertices and redirection probability \(p\).

The digraph is constructed by adding vertices with a link to one previously added vertex. The vertex to link to is chosen uniformly. With probability p, the arc is instead redirected to the successor vertex. The digraph is always a tree. See [KR2001b] for more details.

INPUT:

• n – integer; number of vertices

• p – redirection probability

• seed – a random.Random seed or a Python int for the random number generator (default: None)

EXAMPLES:

sage: # needs networkx
sage: D = digraphs.RandomDirectedGNR(25, .2)
sage: D.is_directed_acyclic()
True
sage: D.to_undirected().is_tree()
True
sage: D.show()                      # long time                             # needs sage.plot

RandomSemiComplete(n)

Return a random semi-complete digraph on \(n\) vertices.

A directed graph \(G=(V,E)\) is semi-complete if for any pair of vertices \(u\) and \(v\), there is at least one arc between them.

To generate randomly a semi-complete digraph, we have to ensure, for any pair of distinct vertices \(u\) and \(v\), that with probability \(1/3\) we have only arc \(uv\), with probability \(1/3\) we have only arc \(vu\), and with probability \(1/3\) we have both arc \(uv\) and arc \(vu\). We do so by selecting a random integer \(coin\) in \([1,3]\). When \(coin==1\) we select only arc \(uv\), when \(coin==3\) we select only arc \(vu\), and when \(coin==2\) we select both arcs. In other words, we select arc \(uv\) when \(coin\leq 2\) and arc \(vu\) when \(coin\geq 2\).

INPUT:

• n – integer; the number of nodes

See also

• Complete()

• RandomTournament()

EXAMPLES:

sage: SC = digraphs.RandomSemiComplete(10); SC
Random Semi-Complete digraph: Digraph on 10 vertices
sage: SC.size() >= binomial(10, 2)                                          # needs sage.symbolic
True
sage: digraphs.RandomSemiComplete(-1)
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

RandomTournament(n)

Return a random tournament on \(n\) vertices.

For every pair of vertices, the tournament has an edge from \(i\) to \(j\) with probability \(1/2\), otherwise it has an edge from \(j\) to \(i\).

INPUT:

• n – integer; number of vertices

EXAMPLES:

sage: T = digraphs.RandomTournament(10); T
Random Tournament: Digraph on 10 vertices
sage: T.size() == binomial(10, 2)                                           # needs sage.symbolic
True
sage: T.is_tournament()
True
sage: digraphs.RandomTournament(-1)
Traceback (most recent call last):
...
ValueError: the number of vertices cannot be strictly negative

See also

• Wikipedia article Tournament_(graph_theory)

• is_tournament()

• TransitiveTournament()

• Complete()

• RandomSemiComplete()

StronglyRegular(n)

Return a Strongly Regular digraph with \(n\) vertices.

The adjacency matrix of the graph is constructed from a skew Hadamard matrix of order \(n+1\). These graphs were first constructed in [Duv1988].

INPUT:

• n – integer, the number of vertices of the digraph.

See also

• sage.combinat.matrices.hadamard_matrix.skew_hadamard_matrix()

• Paley()

EXAMPLES:

A Strongly Regular digraph satisfies the condition \(AJ = JA = kJ\) where \(A\) is the adjacency matrix:

sage: # needs sage.combinat sage.modules
sage: g = digraphs.StronglyRegular(7); g
Strongly regular digraph: Digraph on 7 vertices
sage: A = g.adjacency_matrix()*ones_matrix(7)
sage: B = ones_matrix(7)*g.adjacency_matrix()
sage: A == B == A[0, 0]*ones_matrix(7)
True

TransitiveTournament(n)

Return a transitive tournament on \(n\) vertices.

In this tournament there is an edge from \(i\) to \(j\) if \(i<j\).

See the Wikipedia article Tournament_(graph_theory) for more information.

INPUT:

• n – integer; number of vertices in the tournament

EXAMPLES:

sage: g = digraphs.TransitiveTournament(5)
sage: g.vertices(sort=True)
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()                                  # needs sage.groups
1

See also

• Wikipedia article Tournament_(graph_theory)

• is_tournament()

• is_transitive()

• RandomTournament()

nauty_directg(graphs, options='', debug=False)

Return an iterator yielding digraphs using nauty’s directg program.

Description from directg –help: Read undirected graphs and orient their edges in all possible ways. Edges can be oriented in either or both directions (3 possibilities). Isomorphic directed graphs derived from the same input are suppressed. If the input graphs are non-isomorphic then the output graphs are also.

INPUT:

• graphs – a Graph or an iterable containing Graph. The graph6 string of these graphs is used as an input for directg.

• options (str) – a string passed to directg as if it was run at a system command line. Available options from directg –help:

  -e<int> | -e<int>:<int>  specify a value or range of the total number of arcs
  -o       orient each edge in only one direction, never both
  -a       only make acyclic orientations (implies -o)
  -f<int>  Use only the subgroup that fixes the first <int> vertices setwise
  -V       only output graphs with nontrivial groups (including exchange of
           isolated vertices).  The -f option is respected.
  -s<int>/<int>  Make only a fraction of the orientations: The first integer is
           the part number (first is 0) and the second is the number of
           parts. Splitting is done per input graph independently.
  

• debug (boolean) – default: False - if True directg standard error and standard output are displayed.

EXAMPLES:

sage: gen = graphs.nauty_geng("-c 3")
sage: dgs = list(digraphs.nauty_directg(gen))
sage: len(dgs)
13
sage: dgs[0]
Digraph on 3 vertices
sage: dgs[0]._bit_vector()
'001001000'
sage: len(list(digraphs.nauty_directg(graphs.PetersenGraph(), options="-o")))
324

Generate non-isomorphic acyclic orientations:

sage: K = graphs.CompleteGraph(4)
sage: all(d.is_directed_acyclic() for d in digraphs.nauty_directg(K, options='-a'))
True
sage: sum(1 for _ in digraphs.nauty_directg(K, options='-a'))
1
sage: S = graphs.StarGraph(4)
sage: all(d.is_directed_acyclic() for d in digraphs.nauty_directg(S, options='-a'))
True
sage: sum(1 for _ in digraphs.nauty_directg(S, options='-a'))
5

See also

• orientations()

• strong_orientation()

• strong_orientations_iterator()

• random_orientation()

tournaments_nauty(n, min_out_degree=None, max_out_degree=None, strongly_connected=False, debug=False, options='')

Iterator over all tournaments on \(n\) vertices using Nauty.

INPUT:

• n – integer; number of vertices

• min_out_degree, max_out_degree – integers; if set to None (default), then the min/max out-degree is not constrained

• debug – boolean (default: False); if True the first line of genbg’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.

• options – string; anything else that should be forwarded as input to Nauty’s genbg. See its documentation for more information : http://cs.anu.edu.au/~bdm/nauty/.

EXAMPLES:

sage: for g in digraphs.tournaments_nauty(4):
....:     print(g.edges(sort=True, labels = False))
[(1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(1, 0), (1, 3), (2, 0), (2, 1), (3, 0), (3, 2)]
[(0, 2), (1, 0), (2, 1), (3, 0), (3, 1), (3, 2)]
[(0, 2), (0, 3), (1, 0), (2, 1), (3, 1), (3, 2)]
sage: tournaments = digraphs.tournaments_nauty
sage: [len(list(tournaments(x))) for x in range(1,8)]
[1, 1, 2, 4, 12, 56, 456]
sage: [len(list(tournaments(x, strongly_connected = True))) for x in range(1,9)]
[1, 0, 1, 1, 6, 35, 353, 6008]