Symmetric Graphs and Flag Graphs

Abstract

With any G-symmetric graph Γ admitting a nontrivial G-invariant partition \(\), we may associate a natural “cross-sectional” geometry, namely the 1-design \(\) in which \(\) for \(\) and \(\) if and only if α is adjacent to at least one vertex in C, where \(\) and \(\) is the neighbourhood of B in the quotient graph \(\) of Γ with respect to \(\). In a vast number of cases, the dual 1-design of \(\) contains no repeated blocks, that is, distinct vertices of B are incident in \(\) with distinct subsets of blocks of \(\). The purpose of this paper is to give a general construction of such graphs, and then prove that it produces all of them. In particular, we show that such graphs can be reconstructed from \(\) and the induced action of G on \(\). The construction reveals a close connection between such graphs and certain G-point-transitive and G-block-transitive 1-designs. By using this construction we give a characterization of G-symmetric graphs such that there is at most one edge between any two blocks of \(\). This leads to, in a subsequent paper, a construction of G-symmetric graphs \(\) such that \(\) and each \(\) is incident in \(\) with \(\) vertices of B.