A family of symmetric graphs with complete quotients

A finite graph Gamma is G-symmetric if it admits G as a group of automorphisms acting transitively on V(Gamma) and transitively on the set of ordered pairs of adjacent vertices of Gamma. If V(Gamma) admits a nontrivial G-invariant partition B such that for blocks B, C is an element of B adjacent in the quotient graph Gamma(B) relative to B, exactly one vertex of B has no neighbour in C, then we say that Gamma is an almost multicover of Gamma(B). In this case there arises a natural incidence structure D(Gamma, B) with point set B. If in addition Gamma(B) is a complete graph, then D(Gamma, B) is a (G, 2)-point-transitive and G-block -transitive 2-(vertical bar B vertical bar, m + 1, lambda) design for some m >= 1, and moreover either lambda = 1 or lambda = m + 1. In this paper we classify such graphs in the case when lambda = m + 1; this together with earlier classifications when lambda = 1 gives a complete classification of almost multicovers of complete graphs.