Vertex-transitive graphs with local action the symmetric group on ordered pairs

TL;DR

This paper investigates finite vertex-transitive graphs where local automorphism groups act as the symmetric group on pairs, proving that the stabilizer of a radius-three ball around any vertex is trivial under these conditions.

Contribution

It establishes a new result linking local symmetric group actions to the triviality of certain stabilizers in vertex-transitive graphs.

Findings

The local action being symmetric on pairs implies triviality of the radius-three stabilizer.

Provides structural insights into vertex-transitive graphs with symmetric local actions.

Enhances understanding of automorphism groups in symmetric graph structures.

Abstract

We consider a finite, connected and simple graph $\Gamma$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(\Gamma)$, the local action $G_x^{\Gamma(x)}$ is the action of $\mathrm{Sym}(n)$ on ordered pairs, we show that the group $G_x^{[3]}$, the pointwise stabiliser of a ball of radius three around $x$, is trivial.