# 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

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.

## 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.

INPUT:

• parameters (boolean) – whether to return the quadruple $$(n, k,\lambda,\mu)$$. If parameters = False (default), this method only returns True and False answers. If parameters=True, the True 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().