Asymptotic Automorphism Groups of Circulant Graphs and Digraphs

TL;DR

This paper investigates the automorphism groups of circulant graphs and digraphs, showing most have minimal automorphism groups, providing counterexamples to a conjecture, and analyzing the asymptotic behavior of non-normal cases.

Contribution

It demonstrates that almost all circulant graphs have minimal automorphism groups and provides counterexamples to a conjecture about normality in Cayley (di)graphs.

Findings

Most circulant graphs have minimal automorphism groups.

Counterexamples to the conjecture that non-small automorphism groups imply normality.

Asymptotic analysis shows no general conclusion for non-normal circulant (di)graphs.

Abstract

We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost all circulant graphs whose order lies in any one of these families, are normal. That almost all Cayley (di)graphs whose automorphism group is not as small as possible are normal was conjectured by the second author, so these results provide counterexamples to this conjecture. It is then shown that there is a large family of integers for which almost every circulant digraph whose order lies in this family and that does not have automorphism group as small as possible, is normal. We additionally explore the asymptotic behavior of the automorphism groups of circulant (di)graphs that are not normal, and show that no general conclusion can be obtained.