2025/06/06 by Junjie Huang, Huang, Jun-Jie · 1 citation
Mathematics · #Finite Group Theory Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2506.05803
For an integer s≥1 and a graph Γ, a path (u0, u1, …, us) of vertices of Γ is called an \em s-geodesic if it is a shortest path from u0 to us. We say that Γ is \em s-geodesic transitive if, for each i≤ s, Γ has at least one i-geodesic, and its automorphism group is transitive on the set of i-geodesics. In 2021, Jin and Praeger [J. Combin. Theory Ser. A 178 (2021) 105349] have studied 3-geodesic transitive graphs of girth 5 or 6, and they also proposed to the problem that to classify s-geodesic transitive graphs of girth 2s-1 or 2s-2 for s=4, 5, 6, 7, 8. The case of s = 4 was investigated in [J. Algebra Combin. 60 (2024) 949--963]. In this paper, we study such graphs with s≥5. More precisely, it is shown that a connected (G,s)-geodesic transitive graph Γ with a nontrivial intransitive normal subgroup N of G which has at least 3 orbits, where G is an automorphism group of Γ and s≥ 5, either Γ is the Foster graph and ΓN is the Tutte's 8-cage, or Γ and ΓN have the same girth and ΓN is (G/N,s)-geodesic transitive. Moreover, it is proved that if G acts quasiprimitively on its vertex set, then G is an almost simple group, and if G acts biquasiprimitively, the stabilizer of biparts of Γ in G is an almost simple quasiprimitive group on each of biparts. In addition, G cannot be primitive or biprimitive.