Finite 2-geodesic transitive graphs of prime valency

TL;DR

This paper classifies certain prime valency graphs with highly symmetric automorphism groups, showing they are either 2-arc transitive or specific antipodal covers of complete graphs, with detailed group and diameter properties.

Contribution

It provides a complete classification of non-complete prime valency graphs with automorphism groups transitive on arcs and 2-geodesics, identifying their structure and uniqueness.

Findings

Graphs are either 2-arc transitive or have prime valency p ≡ 1 mod 4

For each such prime p, there is a unique non-bipartite antipodal double cover of K_{p+1}

Automorphism group is PSL(2,p)×Z_2 with diameter 3

Abstract

We classify non-complete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of $2$-geodesics. We prove that either $\Gamma$ is 2-arc transitive or the valency $p$ satisfies $p\equiv 1\pmod 4$, and for each such prime there is a unique graph with this property: it is a non-bipartite antipodal double cover of the complete graph $K_{p+1}$ with automorphism group $PSL(2,p)\times Z_2$ and diameter 3.