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

Research paper by **Pablo Spiga**

Indexed on: **16 Sep '15**Published on: **16 Sep '15**Published in: **Mathematics - Combinatorics**

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#### Abstract

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