Static dense graphs

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.

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 \((n, k,\lambda,\mu)\). 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().