A family of symmetric graphs with complete quotients

Abstract

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 ϵ 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 G- block-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..