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

TL;DR

This work delivers a definitive classification of connected distance transitive graphs of diameter $3$ by leveraging almost simple primitive groups of rank $4$ and detailing their subdegrees. It extends Jin and Tan’s results on $4$-geodesic transitive graphs with intransitive normal subgroups through a quotient/covers framework, showing either both the graph and its quotient are known or they share girth properties with geodesic transitivity in the quotient. A comprehensive catalog of diameter-3 distance transitive graphs is provided, including explicit girth-based refinements—identifying examples such as the M$_{23}$-graph, Sylvester graph, and M$_{23}$-coset graphs—along with structural consequences for geodesic transitive graphs. The final part analyzes $4$-geodesic transitive graphs of girth $6$ or $7$, establishing a reduction to quotient graphs and ruling out many potential covers, thereby clarifying when such graphs can occur and when they must be among previously known constructions. Overall, the paper advances the understanding of the interaction between geodesic transitivity, distance transitivity, and the permutation-group structure governing their automorphism actions.

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.

Theorems & Definitions (26)

• Theorem 1.2: JT24
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Example 2.6