Finite 2-geodesic transitive graphs of prime valency

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 $Γ$ 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.

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