Two-Geodesic-Transitive Graphs Which are Neighbor Cubic or Neighbor Tetravalent

Abstract

A vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a \emph{$2$-geodesic} if $u\neq w$ and $u,w$ are not adjacent. A graph $\Gamma$ is said to be \emph{$2$-geodesic-transitive} if its automorphism group is transitive on both arcs and 2-geodesics. In this paper, a complete classification is given of $2$-geodesic-transitive graphs which are neighbor cubic or neighbor tetravalent.