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

Abstract

Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition $\text{B}$. For $\text{B∈}\text{B}$, let $\text{D}\text{(B)=(B,Γ}{}_{\mathrm{<mtext>B</mtext>}}\text{(B),}\text{I}\text{)}$ be the 1-design in which αIC for α∈B and $\text{C∈Γ}{}_{\mathrm{<mtext>B</mtext>}}\text{(B)}$ if and only if α is adjacent to at least one vertex of C, where $\text{Γ}{}_{\mathrm{<mtext>B</mtext>}}\text{(B)}$ is the neighbourhood of B in the quotient graph $\text{Γ}{}_{\mathrm{<mtext>B</mtext>}}$ of Γ relative to $\text{B}$. In a natural way the setwise stabilizer G_B of B in G induces a group of automorphisms of $\text{D}\text{(B)}$. In this paper, we study those graphs Γ such that the actions of G_B on B and $\text{Γ}{}_{\mathrm{<mtext>B</mtext>}}\text{(B)}$ are permutationally equivalent, that is, there exists a bijection $\text{ρ}\text{:}\text{B→Γ}{}_{\mathrm{<mtext>B</mtext>}}\text{(B)}$ 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 $\text{B}$. By using this labelling technique we analyse $\text{Γ}{}_{\mathrm{<mtext>B</mtext>}}\text{,}\text{D}\text{(B)}$ and the bipartite subgraph Γ[B,C] induced by adjacent blocks B,C of $\text{B}$, 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 $\text{Γ}{}_{\mathrm{<mtext>B</mtext>}}$ via this construction.