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 (B) = (B,ΓA(B),I) be the 1-design in which αIC for α∈B and C∈Γℬ(B) if and only if α is adjacent to at least one vertex of C, where ΓA(B) is the neighbourhood of B in the quotient graph ΓA of Γ relative to ℬ. In a natural way the setwise stabilizer GB of B in G induces a group of automorphisms of (B). In this paper, we study those graphs Γ such that the actions of GB on B and ΓA(B) are permutationally equivalent, that is, there exists a bijection ρ : B → ΓA(B) such that ρ(αx)) = (ρ(α))x for α ∈ B and x ∈ GB. In this case the vertices of Γ can be labelled naturally by the arcs of ℬ. By using this labell..