Finite symmetric graphs with two-arc transitive quotients II

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 Γ relative to B can be (G, 2)-arc transitive even if Γ 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 ΓB is (G, 2)-arc transitive and, for blocks B, C eB adjacent in ΓB, there are exactly |B| - 2(≥1) vertices in B 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 eB) by analyzing the dual D*(B) of the 1-design D(B):=(B,ΓB(B),I), where ΓB(B) is the neighborhood of B in ΓB and αIC (α e B, C e ΓB(B)* in D(B) if and only if α has at least one neighbor in C. In this case, a crucial feature is that D*(B) admits G as a group of automorphisms acting 2-transitively on points and transitively on blocks and flags. It is proved that the case when no point of D(B) is incident with two blocks can be reduced to multicovers, and the case when no point of D(B) is incident with two blocks can be partially reduced to the 3-arc graph construction, where D(B) is the complement of D(B). In the general situation, both D*(B) and its complement D* are (G, 2)-point-transitive and G-block-transitive 2-designs, and exploring relationships between them and Γ is an attractive research direction. In the article we investigate the degenerate case where D*(B) or D* is a trivial Steiner system with block size 2, that is, a complete graph. In each of these cases, we give a construction which produces symmetric graphs with the corresponding properties, and we prove further that every such graph Γ can be constructed from ΓB by using the construction. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 167193, 2007