Tetravalent arc-transitive graphs with unbounded vertex-stabilisers

TL;DR

This paper introduces two new infinite families of tetravalent arc-transitive graphs with unbounded vertex-stabilisers, expanding the known examples beyond the classical cases and contributing to the understanding of symmetry in such graphs.

Contribution

It presents two novel infinite families of tetravalent arc-transitive graphs with large vertex-stabilisers, enriching the classification of symmetric graphs.

Findings

Two new infinite families of graphs with large vertex-stabilisers

Expansion of known examples beyond classical cases

Enhanced understanding of symmetry properties in tetravalent graphs

Abstract

It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilisers, each significant for its own reason.