Two-geodesic transitive graphs of order pn with n ≤ 3

A vertex triple $(u,v,w)$ of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive if its automorphism group is transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order $p^n$ is given for each prime p and $n\le 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph $K_{4,4}$ of order 8, the Schläfli graph of order 27 and its complement, and fourteen infinite families: the cycles $C_p,C_{p^2}$ and $C_{p^3}$, the complete graphs $K_p,K_{p^2}$ and $K_{p^3}$, the complete multipartite graphs $K_{p\left[p\right]}$, $K_{p\left[p^2\right]}$ and $K_{p^2\left[p\right]}$, the Hamming graph $H(2,p)$ and its complement, the Hamming graph $H(3,p)$, and two infinite families of normal Cayley graphs on the extraspecial group of order $p^3$ and exponent p.