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

Let Gamma be a finite G-symmetric graph whose vertex set admits a non-trivi al G-invariant partition B with block size v. A framework for studying such graphs Gamma was developed by Gardiner and Praeger which involved an analy sis of the quotient graph Gamma(B) relative to B, the bipartite subgraph Ga mma[B, C] of Gamma induced by adjacent blocks B, C of Gamma(B) and a certai n 1-design D(B) induced by a block B is an element of 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 greater than or equal to 2, the setw ise stabilizer G(B) is doubly transitive on B and G is faithful on B. We pr ove that D(B) contains no repeated blocks if and only if Gamma(B) is (G, 2) -arc transitive and give a method for constructing such a graph from a 2-ar c 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-a rc transitive graph, and every S-arc transitive graph of even valency, may occur as Gamma(B) for some graph Gamma with these properties. We prove furt her that Gamma[B, C] congruent to K-v-1,K-v-1 if and only if Gamma(B) is (G , 3)-arc transitive.