A class of finite symmetric graphs with 2 - arc transitive quotients

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial Ginvariant partition B with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph ΓB relative to B, the bipartite subgraph Γ[B,C] of Γ induced by adjacent blocks B,C of ΓB and a certain 1-design D(B) induced by a block B ∈ B. The present paper studies the case where the size k of the blocks of D(B) satisfies k = v − 1. In the general case, where k = v − 1 > 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on B. We prove that D(B) contains no repeated blocks if and only if ΓB is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as ΓB for some graph Γ with these properties. We prove further that Γ[B,C]%Kv−1,v−1 if and only if ΓB is (G, 3)-arc transitive.