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

### Primož Potočnik Univerza v Ljubljani

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Minisymposium: APPLICATIONS OF GROUPS IN GRAPH THEORY

Content: Given a finite connected graph $\Gamma$ and a group $G$ acting transitively on the arcs of $\Gamma$, one defines the {\em local group} $G_v^{\Gamma(v)}$ as the permutation group induced by the action of the stabiliser $G_v$ on the neighbourhood $\Gamma(v)$ of a vertex $v$. I will address the following question: To what extent does (the isomorphism class of) $G_v^{\Gamma(v)}$ bound the order of $G_v$? More precisely, I will present some examples of local groups $G_v^{\Gamma(v)}$ for which one can bound the order of $G_v$ by a polynomial function (but not constant!) of the order of the graph, and some examples where this is not possible. To put this in a historical context, the celebrated Tutte's result shows that for $G_v^{\Gamma(v)} \cong {\rm Sym(3)}$ one can bound $|G_v|$ by the constant function $f(n) = 48$. Most of the original results presented in this talk have appeared in the paper by Gabriel Verret, Pablo Spiga and myself published last year in volume 366 of the Transactions of AMS under the same title as this talk.

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