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.