On Automorphisms and Structural Properties of Generalized Cayley Graphs

Abstract

In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order $2p$ is a Cayley graph, where $p$ is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is proved that every generalized Cayley graph on an Abelian group with respect to an automorphism which acts as inversion is a Cayley graph if and only if the group is elementary Abelian $2$-group, or its Sylow $2$-subgroup is cyclic. Necessary and sufficient conditions for a generalized Cayley graph to be unworthy are given.