On the isomorphisms and automorphism groups of circulants

Abstract

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 modulo n, and $$\tilde M = \left\{ {\left. {m, - m} \right|m \in M} \right\}$$ In this paper, we discuss the problems about the automorphism group and isomorphisms of C n (S). When $$M \subseteq S \subseteq \tilde M$$ , we determine the automorphism group of C n (S) and prove that for any T ⊆ Z n , C n (S) ? C n (T) if and only if T = ?S, where ? 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.