On the order of vertex-stabilisers in vertex-transitive graphs with local group ▫${\rm C}_p\times {\rm C}_p$▫ or ▫${\rm C}_{p}{\rm wr}\,{\rm C}_2$▫
Spiga, Pablo ; Verret, Gabriel
Let ▫$p$▫ be a prime and let ▫$\mathcal{L}$▫ be either the intransitive permutation group ▫$\mathrm{C}_p \times \mathrm{C}_p$▫ of degree ▫$2p$▫ or the transitive permutation group ▫$\mathrm{C}_p\, ...\mathrm{wr}\, \mathrm{C}_2$▫ of degree ▫$2p$▫. Let ▫$\Gamma$▫ be a connected ▫$G$▫-vertex-transitive and ▫$G$▫-edge-transitive graph and let ▫$v$▫ be a vertex of ▫$\Gamma$▫. We show that if the permutation group induced by the vertex-stabiliser ▫$G_v$▫ on the neighbourhood ▫$\Gamma(v)$▫ is isomorphic to ▫$\mathcal{L}$▫ then either ▫$|\mathrm{V}(\Gamma)| \geq p |G_v| \log_p (| G_v | / 2)$▫, or ▫$|\mathrm{V}(\Gamma)|$▫ is bounded by a constant depending only on ▫$p$▫, or ▫$\Gamma$▫ is a very-well understood graph. This generalises a few recent results.
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