Abstract :
Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition B. For B∈B, let D(B)=(B,ΓB(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 ΓB(B) is the neighbourhood of B in the quotient graph ΓB of Γ relative to B. In a natural way the setwise stabilizer GB 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 GB on B and ΓB(B) are permutationally equivalent, that is, there exists a bijection ρ : B→ΓB(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 labelling technique we analyse ΓB, 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 ΓB via this construction.
