Critical groups of van Lint-Schrijver Cyclotomic Strongly Regular Graphs

The \emph{critical} group of a finite connected graph is an abelian group defined by the Smith normal form of its Laplacian. Let \(q\) be a power of a prime and \(H\) be a multiplicative subgroup of \(K=\mathbb{F}_{q}\). By \(\mathrm{Cay}(K,H)\) we denote the Cayley graph on the additive group of \(K\) with `connection' set \(H\). A strongly regular graph of the form \(\mathrm{Cay}(K,H)\) is called a \emph{cyclotomic strongly regular graph}. Let \(p\) and \(\ell >2\) be primes such that \(p\) is primitive \(\pmod{\ell}\). We compute the \emph{critical} groups of a family of \emph{cyclotomic strongly regular graphs} for which \(q=p^{(\ell-1)t}\) (with \(t\in \mathbb{N}\)) and \(H\) is the unique multiplicative subgroup of order \(k=\frac{q-1}{\ell}\). These graphs were first discovered by van Lint and Schrijver in \cite{VS}.