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