Finite symmetric graphs with two-arc transitive quotients II

Let Gamma be a finite G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B. It was observed that the quotient graph Gamma(B), of Gamma relative to B can be (G, 2)-arc transitive even if Gamma 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 (17, B) such that Gamma(B) is (G, 2)-arc transitive and, for blocks B, C E B adjacent in Gamma(B), there are exactly I vertical bar B vertical bar - 2 (>= 1) vertices in B which have neighbors in C. In the present article we investigate the general case where 1713 is (G, 2)-arc transitive and is not multicovered by Gamma (i.e., at least one vertex in B has no neighbor in C for adjacent B, C is an element of B) by analyzing the dual D*(B) of the 1-design D(B) := (B, Gamma(B)(B), 1), where Gamma(B)(B) is the neighborhood of B in Gamma(B) and oil C (alpha E B, C E 1713(B)) in D(B) if and only if alpha 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 15(B) is incident with two blocks can be partially reduced to the 3-arc graph construction, where (D) over bar (B) is the complement of D(B). In the general situation, both D*(B) and its complement V(B) are (G, 2)-point-transitive and G-block-transitive 2-designs, and exploring relationships between them and Gamma is an attractive research direction. In the article we investigate the degenerate case where D*(B) or (D) over bar*(B) 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 Gamma can be constructed from Gamma(B) by using the construction. (c) 2007 Wiley Periodicals, Inc.