Automorphisms of Cayley graphs on generalised dicyclic groups

TL;DR

This paper investigates the automorphism groups of Cayley graphs on generalized dicyclic groups, showing that almost all such graphs have automorphism groups as small as possible, similar to abelian groups.

Contribution

It extends known results about automorphism groups of Cayley graphs from abelian groups to generalized dicyclic groups, identifying that most such graphs have minimal automorphism groups.

Findings

Almost all Cayley graphs on generalized dicyclic groups have only the obvious automorphisms.

The automorphism group of these graphs is as small as possible for most cases.

The results parallel those previously known for abelian groups.

Abstract

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on such a group admits specific additional graph automorphisms that depend only on the group. Recently, Dobson and the last two authors showed that almost all Cayley graphs on abelian groups admit no automorphisms other than these obvious necessary ones. In this paper, we prove the analogous result for Cayley graphs on the remaining family of exceptional groups: generalised dicyclic groups.