An application of the Local C(G,T) Theorem to a conjecture of Weiss

2015/09/16 by Pablo Spiga, Spiga, Pablo · 2 citations
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1509.04862

openalex publication_date 2015/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be a connected G-vertex-transitive graph, let v be a vertex of Γ and let G_v^Γ(v) be the permutation group induced by the action of the vertex-stabiliser G_v on the neighbourhood Γ(v). The graph Γ is said to be G-locally primitive if G_v^Γ(v) is primitive. Richard Weiss conjectured in 1978 that, there exists a function f:ℕ→ ℕ such that, if Γ is a connected G-vertex-transitive locally primitive graph of valency d and v is a vertex of Γ with |G_v| finite, then |G_v|≤ f(d). As an application of the Local C(G,T) Theorem, we prove this conjecture when G_v^Γ(v) contains an abelian regular subgroup. In fact, we show that the point-wise stabiliser in G of a ball of Γ of radius 4 is the identity subgroup.

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