Locally s-arc-transitive graphs arising from product action

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.