2025/06/05 by Junjie Huang, Huang, Jun-Jie · 2 citations
Mathematics · #05C25 #20B15 #20B30 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2506.04670
openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a graph Γ, the \em distance dΓ(u,v) between two distinct vertices u and v in Γ is defined as the length of the shortest path from u to v, and the \em diameter diam(Γ) of Γ is the maximum distance between u and v for all vertices u and v in the vertex set of Γ. For a positive integer s, a path (u0,u1,…,us) is called an \em s-geodesic if the distance of u0 and us is s. The graph Γ is said to be \em distance transitive if for any vertices u,v,x,y of \Ga such that d_\Ga(u,v)=d_\Ga(x,y), there exists an automorphism of Γ that maps the pair (u,v) to the pair (x,y). Moreover, Γ is said to be \em geodesic transitive if for each i≤ diam(\Ga), the full automorphism group acts transitively on the set of all i-geodesics. In the monograph [Distance-Regular Graphs, Section 7.5], the authors listed all distance transitive graphs of valency at most 13. By using this classification, in this paper, we provide a complete classification of geodesic transitive graphs with valency at most 13. As a result, there are exactly seven graphs of valency at most 13 that are distance transitive but not geodesic transitive.