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The critical groups of the Peisert graphs $P^*(q)$

The critical group of a finite graph is an abelian group defined by the Smith normal form of the Laplacian. We determine the 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 thatthe 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.