Vertex quasiprimitive two-geodesic transitive graphs

TL;DR

This paper classifies and analyzes vertex quasiprimitively 2-geodesic transitive graphs, focusing on their automorphism group actions and providing examples and classifications for specific action types.

Contribution

It determines all possible quasiprimitive action types for these graphs and classifies those with primitive automorphism groups of PA type.

Findings

Identified all quasiprimitive action types for the graphs.

Provided examples for each quasiprimitive type.

Classified 2-geodesic transitive graphs with primitive PA-type automorphism groups.

Abstract

For a non-complete graph $\Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u\neq w$ and $u,w$ are not adjacent. Then $\Gamma$ is said to be $2$-geodesic transitive if its automorphism group is transitive on both arcs and 2-geodesics. In previous work the author showed that if a $2$-geodesic transitive graph $\Gamma$ is locally disconnected and its automorphism group $\Aut(\Gamma)$ has a non-trivial normal subgroup which is intransitive on the vertex set of $\Gamma$, then $\Gamma$ is a cover of a smaller 2-geodesic transitive graph. Thus the `basic' graphs to study are those for which $\Aut(\Gamma)$ acts quasiprimitively on the vertex set. In this paper, we study 2-geodesic transitive graphs which are locally disconnected and $\Aut(\Gamma)$ acts quasiprimitively on the vertex set. We first determine all the possible quasiprimitive action types and give examples for them, and then classify the family of $2$-geodesic transitive graphs whose automorphism group is primitive on its vertex set of $\PA$ type.