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}.
Categories: math.CO