2025/12/26 by Jun-Jie Huang, Huang, Jun-Jie
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2512.22013
openalex publication_date 2025/12/26 · openalex created_date 2025/12/30 · openalex updated_date 2026/07/28
For an integer s≥1 and a graph Γ, a path (u0, u1, …, us) composed of vertices of Γ is called an \em s-geodesic if it is a shortest path between u0 and us. We say that Γ is \em s-geodesic transitive if for each i≤ 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 ΓN are known, or Γ and ΓN have the same girth and ΓN is (G/N,4)-geodesic transitive.