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

Abstract

For an integer $s\ge 1$ and a graph $\hat{I}“$, a path $(u_0,u_1,\dots ,u_s)$ composed of vertices of $\hat{I}“$ is called an {\em $s$-geodesic} if it is a shortest path between $u_0$ and $u_s$. We say that $\hat{I}“$ is {\em $s$-geodesic transitive} if for each $i\le s$, $\hat{I}“$ 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 $\hat{I}“$, it is shown that either both $\hat{I}“$ and $\hat{I}{“}_N$ are known, or $\hat{I}“$ and $\hat{I}{“}_N$ have the same girth and $\hat{I}{“}_N$ is $(G/N,4)$-geodesic transitive.