Isomorphism of circulant graphs and digraphs

Abstract Let S⊆ {1, …, n−1} satisfy −S = S mod n. The circulant graph G(n, S) with vertex set {v0, v1,…, vn−1} and edge set E satisfies vivj ϵ E if and only if j − i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = − S. Adam conjectured that G(n, S) ≊ G(n, S′) if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p2, and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime, or n is divisible by 24.