Finite $2$-Geodesic Transitive Graphs of Valency $3p$

Abstract

For a non-complete graph $\mathrm{\Gamma }$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u\ne w$ and $u,w$ are not adjacent. Then $\mathrm{\Gamma }$ is said to be $2$-geodesic transitive if its automorphism group is transitive on both arcs and $2$-geodesics. In this paper, we classify the family of connected $2$-geodesic transitive graphs of valency $3p$, where $p$ is an odd prime.