Characterising finite locally s-arc transitive graphs with a star normal quotient∗

Let Γ be a finite locally (G, s)-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of the bipartition are G-orbits. In previous work the authors showed that if G has a nontrivial normal subgroup intransitive on both of the vertex orbits of G, then Γ is a cover of a smaller locally s-arc transitive graph. Thus the “basic” graphs to study are those for which G acts quasiprimitively on at least one of the two orbits. In this paper we investigate the case where G is quasiprimitive on only one of the two G-orbits. Such graphs have a normal quotient which is a star. We construct several infinite families of locally 3-arc transitive graphs and prove characterisation results for several of the possible quasiprimitive types for G.