On the isomorphisms and automorphism groups of circulants

Denote by C-n(S) the circulant graph (or digraph). Let M be a minimal generating element subset of Z(n), the cyclic group of integers module n, and (M) over tilde = {m, -m\m is an element of M}. In this paper, we discuss the problems about the automorphism group and isomorphisms of C-n(S). When M subset of or equal to S subset of or equal to (M) over tilde, we determine the automorphism group of C-n(S) and prove that for any T subset of or equal to Z(n), C-n(S) congruent to C-n(T) if and only if T = lambda S, where lambda is an integer relatively prime to n. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.