On distance transitive graphs and $4$-geodesic transitive graphs

Abstract

For an integer $s\geq1$ and a graph $Γ$, a path $(u_0, u_1, \ldots, u_{s})$ composed of vertices of $Γ$ is called an {\em $s$-geodesic} if it is a shortest path between $u_0$ and $u_s$. We say that $Γ$ is {\em $s$-geodesic transitive} if for each $i\leq s$, $Γ$ contains at least one $i$-geodesic, and its automorphism group acts transitively on the set of all $i$-geodesics. In this paper, by using the classification of almost simple primitive groups of rank $4$, we first classify all distance transitive graphs of diameter $3$. The resulting classification encompasses $73$ classes of graphs. As an application of this result, we have extended the main result of Jin and Tan [J. Algebra Combin. 60 (2024) 949--963]. More precisely, for a connected $(G,4)$-geodesic transitive graph with a nontrivial intransitive normal subgroup $N$ of $G$ that has at least $3$ orbits, where $G$ is an automorphism group of $Γ$, it is shown that either both $Γ$ and $Γ_N$ are known, or $Γ$ and $Γ_N$ have the same girth and $Γ_N$ is $(G/N,4)$-geodesic transitive.