Determining the vertex stabilizers of 4-valent half-arc-transitive graphs

TL;DR

This paper develops a new framework to identify vertex stabilizers in 4-valent half-arc-transitive graphs, significantly expanding the known list and confirming a conjecture about specific stabilizers.

Contribution

It introduces a general theory for determining when concentric groups are 4-HAT-stabilizers, extending the classification of such groups.

Findings

Extended the list of known 4-HAT-stabilizers.

Confirmed that al imes C_2^{m-7} are 4-HAT-stabilizers for m .

Provided a framework to determine 4-HAT-stabilizers.

Abstract

We say that a group is a $4$-HAT-stabilizer if it is the vertex stabilizer of some connected $4$-valent half-arc-transitive graph. In 2001, Maru\v{s}i\v{c} and Nedela proved that every $4$-HAT-stabilizer must be a concentric group. However, over the past two decades, only a very small proportion of concentric groups have been shown to be $4$-HAT-stabilizers. This paper develops a theory that provides a general framework for determining whether a concentric group is a $4$-HAT-stabilizer. With this approach, we significantly extend the known list of $4$-HAT-stabilizers. As a corollary, we confirm that $\mathcal{H}_7\times C_2^{m-7}$ are $4$-HAT-stabilizers for $m\geq 7$, achieving the goal of a conjecture posed by Spiga and Xia.