Static dense graphs¶
This module gathers everything which is related to static dense graphs, i.e. :
- The vertices are integer from \(0\) to \(n-1\)
- 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 |
Fills a binary matrix with the information of a (di)graph. |
Python functions
is_strongly_regular() |
Tests if a graph is strongly regular |
Functions¶
-
sage.graphs.base.static_dense_graph.
is_strongly_regular
(g, parameters=False)¶ Tests whether
self
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 non-adjacent 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 Wikipedia article Strongly regular graph
INPUT:
parameters
(boolean) – 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)]).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)]).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.
triangles_count
(G)¶ Return the number of triangles containing \(v\), for every \(v\).
INPUT:
G
– a simple graph
EXAMPLE:
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