Imprimitive symmetric graphs, 3-arc graphs and 1-designs

Abstract

Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition B . For B∈ B , let D(B)=(B,Γ B (B), I) be the 1-design in which αI C for α∈ B and C∈Γ B (B) if and only if α is adjacent to at least one vertex of C, where Γ B (B) is the neighbourhood of B in the quotient graph Γ B of Γ relative to B . In a natural way the setwise stabilizer G B of B in G induces a group of automorphisms of D(B) . In this paper, we study those graphs Γ such that the actions of G B on B and Γ B (B) are permutationally equivalent, that is, there exists a bijection ρ : B→Γ B (B) such that ρ( α x )=( ρ( α)) x for α∈ B and x∈ G B . In this case the vertices of Γ can be labelled naturally by the arcs of B . By using this labelling technique we analyse Γ B , D(B) and the bipartite subgraph Γ[ B, C] induced by adjacent blocks B, C of B , and study the influence of them on the structure of Γ. We prove that the class of such graphs Γ is precisely the class of those graphs obtained from G-symmetric graphs Σ and self-paired G-orbits on 3-arcs of Σ by a construction introduced in a recent paper of Li, Praeger and the author, and that Γ can be reconstructed from Γ B via this construction.