On automorphism groups of half-arc-transitive tetravalent graphs

TL;DR

This paper characterizes certain automorphism groups of tetravalent graphs, constructs examples including counterexamples to previous questions, and introduces a novel non-normal Cayley graph on a nonabelian simple group.

Contribution

It provides a complete characterization of automorphism groups for half-arc-transitive tetravalent graphs and constructs new examples, including counterexamples and a unique Cayley graph.

Findings

Counterexample to a 2019 question by Rivera and Šparl

First tetravalent non-normal Cayley graph on a nonabelian simple group

Explicit constructions of graphs with specified automorphism properties

Abstract

We characterize connected tetravalent graphs $\Gamma$ which admit groups $M<H$ of automorphisms such that $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive. Examples for each case are constructed, including a counter-example to a question asked by A. R. Rivera and P. \v{S}parl in 2019 as well as the first example of tetravalent normal-edge-transitive non-normal Cayley graph on a nonabelian simple group.