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

TL;DR

This paper determines the critical groups of Peisert graphs, a family of strongly regular graphs, and explores their spectral and Smith normal form properties over finite fields and local integers.

Contribution

It explicitly computes the critical groups of Peisert graphs and shows their adjacency matrices are similar and cospectral over certain fields, revealing new algebraic properties.

Findings

Critical groups of Peisert graphs are explicitly determined.

Adjacency matrices over fields of order p^2 are similar and cospectral.

Pairs of graphs exhibit Smith normal form equivalence and spectral similarity.

Abstract

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.