Two-geodesic transitive graphs of valency six

For a positive integer s less than or equal to the diameter of a graph Γ, an s-geodesic of Γ is a path (v0,v1,…,vs) such that the distance between v0 and vs is s. The graph Γ is said to be s-geodesic transitive, if Γ contains an s-geodesic and its automorphism group is transitive on the set of t-geodesics for all t≤s. In particular, if Γ is s-geodesic transitive with s equal to the diameter of Γ, then Γ is called geodesic transitive. In this paper, we classify the family of finite 2-geodesic transitive graphs of valency 6. Then we completely determine such graphs which are geodesic transitive.