Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

The main result of this paper is that, if Γ is a connected 4-valent G-arc-transitive graph and v is a vertex of Γ, then either Γ   is part of a well-understood infinite family of graphs, or |Gv|≤2436|Gv|≤2436 or 2|Gv|log2⁡(|Gv|/2)≤|VΓ|2|Gv|log2⁡(|Gv|/2)≤|VΓ| and that this last bound is tight. As a corollary, we get a similar result for 3-valent vertex-transitive graphs.