Locally s-arc-transitive graphs arising from product action

TL;DR

This paper investigates locally s-arc-transitive graphs with quasiprimitive product action, proving certain non-primitivity properties and constructing the first examples of such graphs that are not standard double covers, advancing understanding of their structure.

Contribution

It establishes non-primitivity of the acting group in specific locally 2-arc-transitive graphs and constructs novel examples of PA type graphs that are not standard double covers.

Findings

Proves the group does not act primitively on either orbit in certain graphs.

Constructs the first examples of non-standard double cover PA type graphs.

Answers the existence question for these specific graphs.

Abstract

We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc-transitive graphs of PA type that are not standard double covers of $s$-arc-transitive graphs of PA type, answering the existence question for these graphs.