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

Let be a G-symmetric graph admitting a nontrivial G-invariant partition B. For B∈B, let D(B)= (B; B(B); I) be the 1-design in which IC for ∈B and C ∈ B(B) if and only if is adjacent to at least one vertex of C, where B(B) is the neighbourhood of B in the quotient graph B of 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 such that the actions of GB on B and B(B) are permutationally equivalent, that is, there exists a bijection :B → B(B) such that ( )= ( ( )) for ∈B and x∈GB. In this case the vertices of can be labelled naturally by the arcs of B. By using this labelling technique we analyse B; D(B) and the bipartite subgraph [B; C] induced by adjacent blocks B; C of B, and study the in9uence of them on the structure of . We prove that the class of such graphs is precisely the class of those graphs obtained from G-symmetric graphs and self-paired G-orbits on 3-arcs of by a construction introduced in a recent paper of Li, Praeger and the author, and that can be reconstructed from B via this construction. c © 2002 Elsevier Science B.V. All rights reserved.