Finite symmetric graphs with two-arc transitive quotients II

Abstract

Let Γ be a finite G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B. It was observed that the quotient graph ΓB of r relative to B can be (G, 2)-arc transitive even if r 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 (Γ, B) such that Γs is (G, 2)-arc transitive and, for blocks B, C ∈ B adjacent in ΓB, there are exactly |B| - 2 (≥1) vertices in S which have neighbors in C. In the present article we investigate the general case where ΓB is (G, 2)-arc transitive and is not multicovered by Γ (i.e., at least one vertex in B has no neighbor in C for adjacent B, C ∈ B)..