Affine flag graphs and classification of a family of symmetric graphs with complete quotients

Abstract

A graph Γ is G-symmetric if G is a group of automorphisms of Γ which is transitive 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 Γ B of Γ relative to B, exactly one vertex of B has no neighbour in C, then Γ is called an almost multicover of Γ B . In this case an incidence structure with point set B arises naturally, and it is a (G,2)-point-transitive and G-block-transitive 2-design if in addition Γ B is a complete graph. In this paper we classify all G-symmetric graphs Γ such that (i) B has block size |B|≥3; (ii) Γ B is complete and almost multi-covered by Γ; (iii) the incid..