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
data_structures/binary_matrix.pxd
, 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

Fill a binary matrix with the information from a Sage (di)graph. 
Python functions
Check whether the graph is strongly regular 

Check whether \(G\) is triangle free 

Return the number of triangles containing \(v\), for every \(v\) 

Iterator over the induced connected subgraphs of order at most \(k\) 
Functions#
 sage.graphs.base.static_dense_graph.connected_full_subgraphs(G, edges_only=False, labels=False, min_edges=None, max_edges=None)[source]#
Return an iterator over the connected subgraphs of \(G\) with same vertex set.
This method implements a iterator over the connected subgraphs of the input (di)graph with the same ground set of vertices. That is, it iterates over every subgraph \(H = (V_H, E_H)\) of \(G = (V, E)\) such that \(V_H = V\), \(E_H \subseteq E\) and \(H\) is connected. Hence, this method may yield a huge number of graphs.
When the input (di)graph \(G\) is not connected, this method returns nothing.
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 not allowededges_only
– boolean (default:False
); whether to return (Di)Graph or list of verticeslabels
– boolean (default:False
); whether to return labelled edges or not. This parameter is used only whenedges_only
isTrue
.min_edges
– integer (default:None
); minimum number of edges of reported subgraphs. By default (None
), this lower bound will be set to \(n  1\).max_edges
– integer (default:None
); maximum number of edges of reported subgraphs. By default (None
), this lower bound will be set to the number of edges of the input (di)graph.
Note
Roughly, this method explores all possible subsets of neighbors of each vertex, which represents a huge number of subsets. We have thus chosen to limit the degree of the vertices of the graphs that can be considered, even if the graph has a single connected subgraph (e.g., a tree). It is therefore recommended to call this method on biconnected components, as done in
connected_subgraph_iterator()
.EXAMPLES:
The complete graph of order 3 has 4 connected subgraphs:
sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs sage: G = graphs.CompleteGraph(3) sage: len(list(connected_full_subgraphs(G))) 4
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import connected_full_subgraphs >>> G = graphs.CompleteGraph(Integer(3)) >>> len(list(connected_full_subgraphs(G))) 4
A cycle of order 5 has 6 connected subgraphs:
sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs sage: G = graphs.CycleGraph(5) sage: len(list(connected_full_subgraphs(G))) 6
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import connected_full_subgraphs >>> G = graphs.CycleGraph(Integer(5)) >>> len(list(connected_full_subgraphs(G))) 6
The House graph has 18 connected subgraphs of order 5:
sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs sage: G = graphs.HouseGraph() sage: L = list(connected_full_subgraphs(G)) sage: len(L) 18 sage: all(g.order() == 5 for g in L) True sage: all(g.is_connected() for g in L) True sage: F = frozenset(frozenset(g.edges(sort=False, labels=False)) for g in L) sage: len(F) 18
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import connected_full_subgraphs >>> G = graphs.HouseGraph() >>> L = list(connected_full_subgraphs(G)) >>> len(L) 18 >>> all(g.order() == Integer(5) for g in L) True >>> all(g.is_connected() for g in L) True >>> F = frozenset(frozenset(g.edges(sort=False, labels=False)) for g in L) >>> len(F) 18
Specifying bounds on the number of edges:
sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs sage: G = graphs.HouseGraph() sage: [g.size() for g in connected_full_subgraphs(G)] [6, 5, 5, 5, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4] sage: [g.size() for g in connected_full_subgraphs(G, max_edges=4)] [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4] sage: [g.size() for g in connected_full_subgraphs(G, min_edges=6)] [6] sage: [g.size() for g in connected_full_subgraphs(G, min_edges=5, max_edges=5)] [5, 5, 5, 5, 5, 5]
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import connected_full_subgraphs >>> G = graphs.HouseGraph() >>> [g.size() for g in connected_full_subgraphs(G)] [6, 5, 5, 5, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4] >>> [g.size() for g in connected_full_subgraphs(G, max_edges=Integer(4))] [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4] >>> [g.size() for g in connected_full_subgraphs(G, min_edges=Integer(6))] [6] >>> [g.size() for g in connected_full_subgraphs(G, min_edges=Integer(5), max_edges=Integer(5))] [5, 5, 5, 5, 5, 5]
Asking for edges only:
sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs sage: G = Graph([(0, 1, "01"), (0, 2, "02"), (1, 2, "12")]) sage: it = connected_full_subgraphs(G, edges_only=True) sage: next(it) [(0, 1), (0, 2), (1, 2)] sage: next(it) [(0, 1), (0, 2)] sage: it = connected_full_subgraphs(G, edges_only=True, labels=True) sage: next(it) [(0, 1, '01'), (0, 2, '02'), (1, 2, '12')] sage: next(it) [(0, 1, '01'), (0, 2, '02')]
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import connected_full_subgraphs >>> G = Graph([(Integer(0), Integer(1), "01"), (Integer(0), Integer(2), "02"), (Integer(1), Integer(2), "12")]) >>> it = connected_full_subgraphs(G, edges_only=True) >>> next(it) [(0, 1), (0, 2), (1, 2)] >>> next(it) [(0, 1), (0, 2)] >>> it = connected_full_subgraphs(G, edges_only=True, labels=True) >>> next(it) [(0, 1, '01'), (0, 2, '02'), (1, 2, '12')] >>> next(it) [(0, 1, '01'), (0, 2, '02')]
Subgraphs of a digraph:
sage: from sage.graphs.base.static_dense_graph import connected_full_subgraphs sage: G = digraphs.Complete(2) sage: list(connected_full_subgraphs(G, edges_only=True)) [[(0, 1)], [(1, 0)], [(0, 1), (1, 0)]]
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import connected_full_subgraphs >>> G = digraphs.Complete(Integer(2)) >>> list(connected_full_subgraphs(G, edges_only=True)) [[(0, 1)], [(1, 0)], [(0, 1), (1, 0)]]
 sage.graphs.base.static_dense_graph.connected_subgraph_iterator(G, k=None, vertices_only=False, edges_only=False, labels=False, induced=True, exactly_k=False)[source]#
Return an terator 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. This parameter is ignored wheninduced
isTrue
.edges_only
– boolean (default:False
); whether to return (Di)Graph or list of edges. Whenvertices_only
isTrue
, this parameter is ignored.labels
– boolean (default:False
); whether to return labelled edges or not. This parameter is used only whenvertices_only
isFalse
andedges_only
isTrue
.induced
– boolean (default:True
); whether to return induced connected sub(di)graph only or also noninduced sub(di)graphs. This parameter can be set toFalse
for simple (di)graphs only.exactly_k
– boolean (default:False
);True
if we only return graphs of order \(k\),False
if we return graphs of order at most \(k\).
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=3, vertices_only=True, exactly_k=True)) [[1, 2, 3], [1, 2, 4], [2, 3, 4]] 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: G = graphs.CompleteGraph(3) sage: len(list(G.connected_subgraph_iterator())) 7 sage: len(list(G.connected_subgraph_iterator(vertices_only=True))) 7 sage: len(list(G.connected_subgraph_iterator(edges_only=True))) 7 sage: len(list(G.connected_subgraph_iterator(induced=False))) 10 sage: G = DiGraph([(0, 1), (1, 0), (1, 2), (2, 1)]) sage: len(list(G.connected_subgraph_iterator())) 6 sage: len(list(G.connected_subgraph_iterator(vertices_only=True))) 6 sage: len(list(G.connected_subgraph_iterator(edges_only=True))) 6 sage: len(list(G.connected_subgraph_iterator(induced=False))) 18
>>> from sage.all import * >>> G = DiGraph([(Integer(1), Integer(2)), (Integer(2), Integer(3)), (Integer(3), Integer(4)), (Integer(4), Integer(2))]) >>> 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] >>> 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]] >>> list(G.connected_subgraph_iterator(k=Integer(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] >>> list(G.connected_subgraph_iterator(k=Integer(3), vertices_only=True, exactly_k=True)) [[1, 2, 3], [1, 2, 4], [2, 3, 4]] >>> list(G.connected_subgraph_iterator(k=Integer(2), vertices_only=True)) [[1], [1, 2], [2], [2, 3], [2, 4], [3], [3, 4], [4]] >>> G = DiGraph([(Integer(1), Integer(2)), (Integer(2), Integer(1))]) >>> list(G.connected_subgraph_iterator()) [Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex] >>> list(G.connected_subgraph_iterator(vertices_only=True)) [[1], [1, 2], [2]] >>> G = graphs.CompleteGraph(Integer(3)) >>> len(list(G.connected_subgraph_iterator())) 7 >>> len(list(G.connected_subgraph_iterator(vertices_only=True))) 7 >>> len(list(G.connected_subgraph_iterator(edges_only=True))) 7 >>> len(list(G.connected_subgraph_iterator(induced=False))) 10 >>> G = DiGraph([(Integer(0), Integer(1)), (Integer(1), Integer(0)), (Integer(1), Integer(2)), (Integer(2), Integer(1))]) >>> len(list(G.connected_subgraph_iterator())) 6 >>> len(list(G.connected_subgraph_iterator(vertices_only=True))) 6 >>> len(list(G.connected_subgraph_iterator(edges_only=True))) 6 >>> len(list(G.connected_subgraph_iterator(induced=False))) 18
 sage.graphs.base.static_dense_graph.is_strongly_regular(g, parameters=False)[source]#
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)
>>> from sage.all import * >>> g = graphs.PetersenGraph() >>> g.is_strongly_regular() True >>> 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)
>>> from sage.all import * >>> g = graphs.ClebschGraph() >>> g.is_strongly_regular() True >>> 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
>>> from sage.all import * >>> g = graphs.ChvatalGraph() >>> g.is_strongly_regular() False
Complete graphs are not strongly regular. (Issue #14297)
sage: g = graphs.CompleteGraph(5) sage: g.is_strongly_regular() False
>>> from sage.all import * >>> g = graphs.CompleteGraph(Integer(5)) >>> 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
>>> from sage.all import * >>> g = graphs.CompleteGraph(Integer(5)).complement() >>> g.is_strongly_regular() False
The empty graph is not strongly regular:
sage: g = graphs.EmptyGraph() sage: g.is_strongly_regular() False
>>> from sage.all import * >>> g = graphs.EmptyGraph() >>> 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().
>>> from sage.all import * >>> Graph([(Integer(1),Integer(1)),(Integer(2),Integer(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(). >>> Graph([(Integer(1),Integer(2)),(Integer(1),Integer(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)[source]#
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
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import is_triangle_free >>> is_triangle_free(graphs.PetersenGraph()) True >>> K4 = graphs.CompleteGraph(Integer(4)) >>> is_triangle_free(K4) False >>> b, certif = is_triangle_free(K4, certificate=True) >>> K4.subgraph(certif).is_clique() True
 sage.graphs.base.static_dense_graph.triangles_count(G)[source]#
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) # needs sage.symbolic True
>>> from sage.all import * >>> from sage.graphs.base.static_dense_graph import triangles_count >>> triangles_count(graphs.PetersenGraph()) {0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0} >>> sum(triangles_count(graphs.CompleteGraph(Integer(15))).values()) == Integer(3) * binomial(Integer(15), Integer(3)) # needs sage.symbolic True