Generalized Cayley graphs of complete groups

TL;DR

This paper explores properties of generalized Cayley graphs of complete groups, characterizes their isomorphisms, and classifies restricted GCI-groups, including symmetric groups.

Contribution

It introduces new characterizations of generalized Cayley graphs of complete groups and provides a complete classification of restricted GCI-groups.

Findings

Characterization of isomorphic generalized Cayley graphs

Necessary and sufficient conditions for complete groups to be GCI groups

Complete classification of restricted GCI-groups for symmetric groups

Abstract

A group $G$ is complete group if it satisfies $Z(G)=e$ and $Aut(G)=Inn(G)$. In this paper, on the one hand, we study the basic properties of generalized Cayley graphs and characterize two classes isomorphic generalized generalized Cayley graphs of complete groups. On the other hand, we give the sufficient and necessary conditions of complete group to be $GCI$ group and restricted $GCI$ group. As an application, we complete the classification of restricted $GCI$-groups for symmetric groups.