Two-geodesic transitive graphs of order $p^n$ with $n\leq3$

TL;DR

This paper classifies all 2-geodesic transitive graphs of order p^n for primes p and n ≤ 3, revealing a small set of specific graphs and infinite families with detailed structural properties.

Contribution

It provides a complete classification of 2-geodesic transitive graphs of order p^n for n ≤ 3, including explicit descriptions of all such graphs and families.

Findings

Identifies three small graphs: K_{4,4}, the Schl"afli graph, and its complement.

Describes fourteen infinite families, including cycles, complete graphs, multipartite graphs, and Hamming graphs.

Includes two infinite families of normal Cayley graphs on extraspecial groups.

Abstract

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 $n\leq 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph $K_{4,4}$ of order $8$, the Schl\"{a}fli 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 $K_{p[p]}$, $K_{p[p^2]}$ and $K_{p^2[p]}$, the Hamming graph $H(2,p)$ and its complement, the Hamming graph $H(3,p)$, and two infinite families of normal Cayley graphs on extraspecial group of order $p^3$ and exponent $p$.