Critical groups of van Lint-Schrijver Cyclotomic Strongly Regular Graphs

Abstract

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}.