Finite \(2\)-Geodesic Transitive Graphs of Valency \(3p\)

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 this paper, we classify the family of connected \(2\)-geodesic transitive graphs of valency \(3p\), where \(p\) is an odd prime.