Critical groups of van Lint-Schrijver Cyclotomic Strongly Regular Graphs

Abstract

The 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 = F q . By 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 Cay(K, H) is called a cyclotomic strongly regular graph. Let ℓ > 2 and p be primes such that p is primitive (mod ℓ). We compute the critical groups of a family of cyclotomic strongly regular graphs for which q = p (ℓ−1)t (with t ∈ N) and H is the unique multiplicative subgroup of order k = q−1 ℓ . These graphs were first discovered by van Lint and Schrijver in [24].