Symmetric graphs with projective subconstituents

Abstract:

Let $\Gamma$ be a finite, undirected, connected graph and G a subgroup of ${\operatorname {aut}}(\Gamma )$ acting transitively on the vertex set of $\Gamma$ such that the stabilizer $G(x)$ in G of a vertex x contains a normal subgroup which induces a permutation group on the set of vertices adjacent to x isomorphic to $PSL(n,q)$ with $n \geqslant 3$. A bound for $|G(x)|$ depending only on n and q is shown to exist under certain conditions.