2021/06/23 by Wei Jin, Jin, Wei
Computer Science · Engineering · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2106.12357
openalex publication_date 2021/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a non-complete graph Γ, a vertex triple (u,v,w) with v adjacent to both u and w is called a 2-geodesic if u≠ w and u,w are not adjacent. Then Γ is said to be 2-geodesic transitive if its automorphism group is transitive on both arcs and 2-geodesics. In previous work the author showed that if a 2-geodesic transitive graph Γ is locally disconnected and its automorphism group \Aut(Γ) has a non-trivial normal subgroup which is intransitive on the vertex set of Γ, then Γ is a cover of a smaller 2-geodesic transitive graph. Thus the `basic' graphs to study are those for which \Aut(Γ) acts quasiprimitively on the vertex set. In this paper, we study 2-geodesic transitive graphs which are locally disconnected and \Aut(Γ) acts quasiprimitively on the vertex set. We first determine all the possible quasiprimitive action types and give examples for them, and then classify the family of 2-geodesic transitive graphs whose automorphism group is primitive on its vertex set of \PA type.