On the order of arc-stabilisers in arc-transitive graphs with prescribed local group

Let $\mathrm{\Gamma }$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of $\mathrm{\Gamma }$ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\mathrm{\Gamma }(v)$. We study the problem of bounding $\mathrm{\mid }{G}_{uv}\mathrm{\mid }$ in terms of $L$ and the order of $\mathrm{\Gamma }$.