Finite normal edge-transitive Cayley graphs

Abstract

An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G , each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G . Moreover, given a normal edge-transitive Cayley graph Γ H for a quotient group G / H , necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has Γ H as a homomorphic image, and a method for obtaining all such graphs Γ is given.