Locally s-arc transitive graphs with trivial edge kernels.
John van Bon
Let $\Delta$ be a connected graph, without loops or multiple edges. Let $G$ be
a group of automorphisms acting locally $s$-arc transitively on $\Delta$. The
vertex kernel of a vertex $z$ is the stabilizer of the vertex and all its
neighbors. The edge kernel of an edge is intersection of the vertex kernels of
the two vertices on the edge. We show that if for each vertex $z$ the valency
is at least 3, $|G_z| < \infty$ and all edge kernels are trivial, then $s \leq
3$. This settles an open problem posed by R. Weiss in 1978.
As a consequence we obtain a characterization of locally s-arc transitive
graphs with $s \geq 4$ and trivial edge kernels. They are polygons or Levi
graphs of covers of the Hoffman-Singleton graph.