Let Gamma be a finite G-symmetric graph whose vertex set admits a non-trivial G-invariant partition B with block size v. A framework for studying such graphs Gamma was developed by Gardiner and Praeger which involved an analysis of the quotient graph Gamma(B) relative to B, the bipartite subgraph Gamma[B, C] of Gamma induced by adjacent blocks B, C of Gamma(B) and a certain 1-design D(B) induced by a block B is an element of B. The present paper studies the case where the size k of the blocks of D(B) satisfies k = v - 1. In the general case, where k = v - 1 greater than or equal to 2, the setwise stabilizer G(B) is doubly transitive on B and G is faithful on B. We prove that D(B) contains no repeated blocks if and only if Gamma(B) is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every S-arc transitive graph of even valency, may occur as Gamma(B) for some graph Gamma with these properties. We prove further that Gamma[B, C] congruent to K-v-1,K-v-1 if and only if Gamma(B) is (G, 3)-arc transitive.
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|Publication status||Published - 2000|