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

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