Symmetric Graphs and Flag Graphs

With any G-symmetric graph admitting a nontrivial G-invariant partition B, we may associate a natural ‘‘cross-sectional’’ geometry, namely the 1-design DðBÞ 1⁄4 ðB; BðBÞ; IÞ in which IC for 2B and C 2 BðBÞ if and only if is adjacent to at least one vertex in C, where B2B and BðBÞ is the neighbourhood of B in the quotient graph B of with respect to B. In a vast number of cases, the dual 1-design of DðBÞ contains no repeated blocks, that is, distinct vertices of B are incident in DðBÞ with distinct subsets of blocks of BðBÞ. 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 B and the induced action of G on B. 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 B. This leads to, in a subsequent paper, a construction of G-symmetric graphs ð ;BÞ such that jBj5 3 and each C 2 BðBÞ is incident in DðBÞ with jBj 1 vertices of B. 2000 Mathematics Subject Classification: 05C25, 05B05, 05E99