On transitive permutation groups with exponential graph growth

2025/08/18 by Mitrović, Đorđe, Gabriel Verret, Verret, Gabriel
Mathematics · #05E18 #20B25 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2508.12588

openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be a finite connected graph and G a vertex-transitive group of its automorphisms. The pair (Γ, G) is said to be locally-L if the permutation group induced by the action of the vertex-stabiliser G_v on the set of neighbours of a vertex v in Γ is permutation isomorphic to L. The maximum growth of |G_v| as a function of |VΓ| for locally-L pairs (Γ,G) is called the graph growth of L. We prove that if L is a transitive permutation group on a set Ω admitting a nontrivial block B such that the pointwise stabiliser of Ω∖ B in L is nontrivial, then the graph growth of L is exponential. This generalises several results in the literature on transitive permutation groups with exponential graph growth.

Citations

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Related

• On the order of vertex-stabilisers in vertex-transitive graphs with\n local group C_p\× C_p or C_p wr C₂
• On graph-restrictive permutation groups
• On intransitive graph-restrictive permutation groups
• Three local actions in 6-valent arc-transitive graphs
• Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph