On automorphisms and structural properties of generalized Cayley graphs

TL;DR

This paper investigates the properties of generalized Cayley graphs, proving conditions under which they are equivalent to Cayley graphs, especially focusing on Abelian groups and automorphisms acting as inversion.

Contribution

It establishes new criteria for when generalized Cayley graphs are actually Cayley graphs, particularly on Abelian groups with specific automorphisms.

Findings

Every generalized Cayley graph of order 2p is a Cayley graph.

Generalized Cayley graphs on Abelian groups with inversion automorphisms are Cayley graphs only under specific group conditions.

Conditions for generalized Cayley graphs to be unworthy are characterized.

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.