Sp(6, 2)’s Family, Plots, and Ramsey Numbers

Strongly regular graphs lie on the cusp between highly structured and unstructured. For example, there is a unique strongly regular graph with parameters (36, 10, 4, 2), but there are 32548 non-isomorphic graphs with parameters (36, 15, 6, 6).

Peter Cameron, Random Strongly Regular Graphs?

This a shorter version of this report which I just put on my homepage. But I added more links. I assume that one is familiar with strongly regular graphs (SRGs). One particular SRG, the collinearity graph of Sp(6, 2), has parameters (63, 30, 13, 15). A very simple technique, Godsil-McKay (GM) switching, can generate many non-isomorphic graphs with the same parameters. More specifically, there are probably billions such graphs and I generated 13 505 292 of them. This is the number of graphs which you obtain by applying a certain type of GM switching (i.e. using a bipartition of type 4, 59) at most 5 times to Sp(6,2). Plots of the number of cliques, cocliques, and the size of the autmorphism group are scattered throughout this post.



Constructing Cospectral Graphs

Last week I had a cold and could not do much thinking. So I spent my time making TikZ pictures for an upcoming talk of mine. This talk is on my recent work with Akihiro Munemasa on constructing cospectral strongly regular graphs. I think that the pictures are nice for a blog post, so here we go.

1. The Spectrum of a Graph

Two graphs {\Gamma} and {\overline{\Gamma}} are called cospectral if their adjacency matrices {A} and {\overline{A}} have the same eigenvalues. This is the same as saying that there is an orthogonal matrix {Q} with {\overline{A} = Q^T A Q}. Any permutation matrix is a valid choice for {Q}, but this is not very interesting as then {\Gamma} and {\overline{\Gamma}} are isomorphic. Chris Godsil and Brendan McKay described one of the easiest interesting choices for {Q} in 1982. For a graph {\Gamma} on {v} vertices, a simplified version of their matrix is

\displaystyle Q = \begin{pmatrix} \frac{1}{m} J_{2m} - I_{2m} & 0 \\ 0 & I_{v-2m} \end{pmatrix}.