Finite symmetric graphs with two-arc transitive quotients II

Let be a finite G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B. It was observed that the quotient graph B of relative to B can be (G,2)-arc transitive even if itself is not necessarily (G,2)-arc transitive. In a previous article of Iranmanesh et al., this observation motivated a study of G-symmetric graphs ( ,B) such that B is (G,2)-arc transitive and, for blocks B,C ∈ B adjacent in B, there are exactly |B| − 2 (≥1) vertices in B which have neighbors in C. In the present article we investigate the general case where B is (G,2)-arc transitive and is not multicovered by (i.e., at least one vertex in B has no neighbor in C for adjacent B,C ∈ B) by analyzing the dual D∗(B) of the 1-design Contract grant sponsor: NSF (to Z.L.); Contract grant sponsor: 973 Project (to Z.L.); Contract grant sponsor: PCSIRT (to Z.L.); Contract grant sponsor: Australian Research Council (to S.Z.); Contract grant number: DP0558677; Contract grant sponsor: University of Melbourne (to S.Z.). Journal of Graph Theory © 2007 Wiley Periodicals, Inc.