Critical groups of van Lint-Schrijver Cyclotomic Strongly Regular Graphs

TL;DR

This paper computes the critical groups of a specific family of cyclotomic strongly regular graphs constructed from finite fields, extending understanding of their algebraic and combinatorial properties.

Contribution

It explicitly determines the critical groups for a class of cyclotomic strongly regular graphs defined by prime powers and subgroup structures.

Findings

Critical groups are computed for the specified family of graphs.

Results depend on prime and subgroup parameters.

Provides explicit algebraic structure of the critical groups.

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}.