A Family of Symmetric Graphs with Complete Quotients

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