# Vertex-stabilisers in locally-transitive graphs

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Luke Morgan

The University of Western Australia

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**Minisymposium:**
APPLICATIONS OF GROUPS IN GRAPH THEORY

**Content:**
A graph $\Gamma$ is $G$-locally-transitive, for $G\leqslant \mathrm{Aut}(\Gamma)$, if for each vertex $x$ of $\Gamma$ the vertex-stabiliser $G_x$ acts transitively on the neighbours of $x$ in $\Gamma$. Additionally, $\Gamma$ is weakly locally projective if for each vertex $x$ the stabiliser $G_x$ induces on the set of vertices adjacent to $x$ a doubly transitive action with socle the projective group $L_{n_x}(q_x)$ for an integer $n_x$ and a prime power $q_x$.
The Mathieu group $M_{24}$ and the Held group $He$ are known to arise as automorphism groups of rank three geometries. In turn, these geometries give rise to weakly locally projective graphs. In this talk I will report on joint work with Michael Giudici, Alexander Ivanov and Cheryl Praeger in which we have begun a program characterising some of the remarkable examples of locally-transitive graphs. We begin by characterising the amalgams arising from the weakly locally projective graphs obtained from $M_{24}$ and $He$. This reveals a new amalgam with the alternating group $A_{16}$ as a completion. We are also able to demonstrate how these amalgams form part of a natural infinite family.