Two families of graphs that are Cayley on nonisomorphic groups

TL;DR

This paper constructs two infinite families of graphs that are Cayley graphs on both abelian and nonabelian groups, including the smallest known examples, expanding understanding of Cayley graph representations.

Contribution

It introduces the first known infinite families of graphs that are Cayley on both abelian and nonabelian groups, including minimal examples not previously identified.

Findings

Constructed two infinite families of such graphs.

Included the smallest known examples of these graphs.

Extended the understanding of Cayley graph representations on different group types.

Abstract

A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the groups have order $pq$; when the Cayley graphs are normal; or when the groups are both abelian. In this paper, we construct two infinite families of graphs, each of which is Cayley on an abelian group and a nonabelian group. These families include the smallest examples of such graphs that had not appeared in other results.