On vertex stabilisers in symmetric quintic graphs

TL;DR

This paper classifies all vertex and edge stabilizers in symmetric graphs of valency five by analyzing group actions on a five-valent tree, completing a classification problem in algebraic graph theory.

Contribution

It provides a complete classification of vertex and edge stabilizers in symmetric quintic graphs, building on previous work and introducing a classification of primitive amalgams of degree (5, 2).

Findings

Classification of all locally finite symmetric actions on a 5-valent tree.

Complete list of isomorphism types of vertex and edge stabilizers.

New classification of finite primitive amalgams of degree (5, 2).

Abstract

In this paper we determine all locally finite and symmetric actions of a group on the tree of valency five. As a corollary we complete the classification of the isomorphism types of vertex and edge stabilisers in a group acting symmetrically on a graph of valency five. This builds on work of Weiss and recent work of Zhou and Feng. This depends upon the second result of this paper, the classification of isomorphism types of finite, primitive amalgams of degree (5, 2).