Static dense graphs¶
This module gathers everything which is related to static dense graphs, i.e. :
 The vertices are integer from \(0\) to \(n1\)
 No labels on vertices/edges
 No multiple edges
 No addition/removal of vertices
This being said, it is technically possible to add/remove edges. The data structure does not mind at all.
It is all based on the binary matrix data structure described in
misc/binary_matrix.pxi
, which is almost a copy of the bitset data
structure. The only difference is that it differentiates the rows (the vertices)
instead of storing the whole data in a long bitset, and we can use that.
For an overview of graph data structures in sage, see
overview
.
Index¶
Cython functions
dense_graph_init 
Fill a binary matrix with the information from a Sage (di)graph. 
Python functions
is_strongly_regular() 
Check whether the graph is strongly regular 
is_triangle_free() 
Check whether \(G\) is triangle free 
triangles_count() 
Return the number of triangles containing \(v\), for every \(v\) 
connected_subgraph_iterator() 
Iterator over the induced connected subgraphs of order at most \(k\) 
Functions¶

sage.graphs.base.static_dense_graph.
connected_subgraph_iterator
(G, k=None, vertices_only=False)¶ Iterator over the induced connected subgraphs of order at most \(k\).
This method implements a iterator over the induced connected subgraphs of the input (di)graph. An induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset (Wikipedia article Induced_subgraph).
As for method
sage.graphs.generic_graph.connected_components()
, edge orientation is ignored. Hence, the directed graph with a single arc \(0 \to 1\) is considered connected.INPUT:
G
– aGraph
or aDiGraph
; loops and multiple edges are allowedk
– (optional) integer; maximum order of the connected subgraphs to report; by default, the method iterates over all connected subgraphs (equivalent tok == n
)vertices_only
– boolean (default:False
); whether to return (Di)Graph or list of vertices
EXAMPLES:
sage: G = DiGraph([(1, 2), (2, 3), (3, 4), (4, 2)]) sage: list(G.connected_subgraph_iterator()) [Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 3 vertices, Subgraph of (): Digraph on 4 vertices, Subgraph of (): Digraph on 3 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 3 vertices, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex] sage: list(G.connected_subgraph_iterator(vertices_only=True)) [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]] sage: list(G.connected_subgraph_iterator(k=2)) [Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex] sage: list(G.connected_subgraph_iterator(k=2, vertices_only=True)) [[1], [1, 2], [2], [2, 3], [2, 4], [3], [3, 4], [4]] sage: G = DiGraph([(1, 2), (2, 1)]) sage: list(G.connected_subgraph_iterator()) [Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex] sage: list(G.connected_subgraph_iterator(vertices_only=True)) [[1], [1, 2], [2]]

sage.graphs.base.static_dense_graph.
is_strongly_regular
(g, parameters=False)¶ Check whether the graph is strongly regular.
A simple graph \(G\) is said to be strongly regular with parameters \((n, k, \lambda, \mu)\) if and only if:
 \(G\) has \(n\) vertices
 \(G\) is \(k\)regular
 Any two adjacent vertices of \(G\) have \(\lambda\) common neighbors
 Any two nonadjacent vertices of \(G\) have \(\mu\) common neighbors
By convention, the complete graphs, the graphs with no edges and the empty graph are not strongly regular.
See the Wikipedia article Strongly regular graph.
INPUT:
parameters
– boolean (default:False
); whether to return the quadruple \((n, k, \lambda, \mu)\). Ifparameters = False
(default), this method only returnsTrue
andFalse
answers. Ifparameters = True
, theTrue
answers are replaced by quadruples \((n, k, \lambda, \mu)\). See definition above.
EXAMPLES:
Petersen’s graph is strongly regular:
sage: g = graphs.PetersenGraph() sage: g.is_strongly_regular() True sage: g.is_strongly_regular(parameters=True) (10, 3, 0, 1)
And Clebsch’s graph is too:
sage: g = graphs.ClebschGraph() sage: g.is_strongly_regular() True sage: g.is_strongly_regular(parameters=True) (16, 5, 0, 2)
But Chvatal’s graph is not:
sage: g = graphs.ChvatalGraph() sage: g.is_strongly_regular() False
Complete graphs are not strongly regular. (trac ticket #14297)
sage: g = graphs.CompleteGraph(5) sage: g.is_strongly_regular() False
Completements of complete graphs are not strongly regular:
sage: g = graphs.CompleteGraph(5).complement() sage: g.is_strongly_regular() False
The empty graph is not strongly regular:
sage: g = graphs.EmptyGraph() sage: g.is_strongly_regular() False
If the input graph has loops or multiedges an exception is raised:
sage: Graph([(1,1),(2,2)],loops=True).is_strongly_regular() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with loops. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow loops using allow_loops(). sage: Graph([(1,2),(1,2)],multiedges=True).is_strongly_regular() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges using allow_multiple_edges().

sage.graphs.base.static_dense_graph.
is_triangle_free
(G, certificate=False)¶ Check whether \(G\) is triangle free.
INPUT:
G
– a Sage graphcertificate
– boolean (default:False
); whether to return a triangle if one is found
EXAMPLES:
sage: from sage.graphs.base.static_dense_graph import is_triangle_free sage: is_triangle_free(graphs.PetersenGraph()) True sage: K4 = graphs.CompleteGraph(4) sage: is_triangle_free(K4) False sage: b, certif = is_triangle_free(K4, certificate=True) sage: K4.subgraph(certif).is_clique() True

sage.graphs.base.static_dense_graph.
triangles_count
(G)¶ Return the number of triangles containing \(v\), for every \(v\).
INPUT:
G
– a simple Sage graph
EXAMPLES:
sage: from sage.graphs.base.static_dense_graph import triangles_count sage: triangles_count(graphs.PetersenGraph()) {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0} sage: sum(triangles_count(graphs.CompleteGraph(15)).values()) == 3 * binomial(15, 3) True