The critical groups of the Peisert graphs

Abstract

The critical group of a finite graph is an abelian group defined by the Smith normal form of the Laplacian. We determine the critical groups of the Peisert graphs, a certain family of strongly regular graphs similar to, but different from, the Paley graphs. It is further shown that the adjacency matrices of the two graphs defined over a field of order $$p^2$$ with $$p\equiv 3\pmod 4$$ are similar over the $$\ell $$ -local integers for every prime $$\ell $$ . Consequently, each such pair of graphs provides an example where all the corresponding generalized adjacency matrices are both cospectral and equivalent in the sense of Smith normal form.