On locally semiprimitive graphs and a theorem of Weiss

TL;DR

This paper studies graphs with specific symmetry properties, establishing bounds on stabilizer sizes and structural theorems, contributing to understanding the Poto
0dnik-Spiga-Verret Conjecture in algebraic graph theory.

Contribution

It introduces bounds on vertex stabilizers for graphs with semiprimitive local actions and provides a detailed structural analysis, advancing the study of symmetric graphs.

Findings

Bound on vertex stabilizer order depending on valency for coprime cases

Structural theorem for vertex stabilizers in remaining cases

Progress towards the Poto
0dnik-Spiga-Verret Conjecture

Abstract

In this paper we investigate graphs that admit a group acting arc-transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup. This type of semiprimitive group is a generalisation of an affine group. We show that if the graph has valency coprime to six, then there is a bound on the order of the vertex stabilisers depending on the valency alone. We also prove a detailed structure theorem for the vertex stabilisers in the remaining case. This is a contribution to an ongoing project to investigate the validity of the Poto\v{c}nik-Spiga-Verret Conjecture.