On Isomorphisms of Vertex-transitive Graphs

TL;DR

This paper extends the study of graph isomorphisms from Cayley graphs to all vertex-transitive graphs, providing new characterizations, examples, and initiating the study of GI and DGI-groups.

Contribution

It generalizes known results on Cayley graphs to vertex-transitive graphs and introduces the concepts of GI and DGI-groups.

Findings

Provided new characterizations of vertex-transitive graph isomorphisms

Constructed examples including a symmetric non-Cayley non-GI-graph

Initiated the study of GI and DGI-groups

Abstract

The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups.