Cubic vertex-transitive graphs admitting automorphisms of large order

TL;DR

This paper investigates cubic vertex-transitive graphs with automorphisms of large order, revealing they are either multicirculants with small parameters or part of an infinite family with girth six.

Contribution

It extends the classification of symmetric cubic graphs by analyzing those with automorphisms of order at least one-third of the graph's size, including non-semiregular cases.

Findings

Graphs are either small multicirculants or belong to an infinite girth 6 family.

Automorphisms of large order impose strong structural constraints.

New classifications extend previous results on symmetric multicirculants.

Abstract

A connected graph of order $n$ admitting a semiregular automorphism of order $n/k$ is called a $k$-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper we study the broader class of cubic vertex-transitive graphs of order $n$ admitting an automorphism of order $n/3$ or larger that may not be semiregular. In particular, we show that any such graph is either a $k$-multicirculant for some $k \leq 3$, or it belongs to an infinite family of graphs of girth $6$.