Arc transitive circulants

TL;DR

This paper characterizes normal arc-transitive circulants and arc-transitive normal circulants, completing their classification by analyzing automorphism groups and element orders within the Cayley graph structure.

Contribution

It provides new characterizations of normal arc-transitive circulants and completes their classification by Li-Xia-Zhou through automorphism group analysis.

Findings

Aut(C,S) is transitive on S iff each element of S has order n

Aut(Γ) contains C iff S does not contain a coset of any subgroup

Complete classification of arc-transitive circulants

Abstract

This short paper presents characterisations of normal arc-transitive circulants and arc-transitive normal circulants, that is, for a connected arc-transitive circulant $\Gamma=\Cay(C,S)$, it is shown that 1. Aut(C,S) is transitive on S if and only if each element of S has order n; 2. $Aut\Gamma\rhd C$ if and only if S does not contain a coset of any subgroup. This completes the classification of arc-transitive circulants given by Li-Xia-Zhou.