© 2014 Elsevier Inc. 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 or 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.
Potočnik, P., Spiga, P., & Verret, G. (2015). Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs. Journal of Combinatorial Theory. Series B, 111, 148-180. https://doi.org/10.1016/j.jctb.2014.10.002