A Construction for Vertex-transitive Graphs

1. Introduction. A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group T in the family all graphs G whose vertex-set is the orbit V of T and for which T ^ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of cyclic groups ((01.. .n— 1)) generated by cycles (01...»-1) of length n, then the corresponding graphs are the ^-vertex circulant graphs. In this paper we consider transitive permutation groups of degree mn generated by a "rotation" p which is a product of m disjoint cycles of length n and by a "twisted translation" r such that rpr~ l = p a for some a. The abstract groups isomorphic to the groups T = (p, T) are the semi-direct products of two cyclic groups. We call the corresponding graphs metacirculants. We note that similar constructions apply to vertex-transitive digraphs, though we shall restrict our attention here to graphs.