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

2025/02/18 by Xia, Binzhou, Zhang, Zhishuo, Zhou, Sanming
#05C25 #20B25 #20B35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2502.12763

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, Marušič 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 H₇× C₂^m-7 are 4-HAT-stabilizers for m≥ 7, achieving the goal of a conjecture posed by Spiga and Xia.

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