Imprimitive Distance-Transitive Graphs with Primitive Core of Diameter at Least 3

A distance-transitive graph G is one upon which the automorphism group acts transitively on ordered pairs of vertices at every fixed distance. Only connected graphs need to be considered. Those of diameter 2 are the rank-3 graphs, whose careful study was initiated by Donald G. Higman in his breakthrough paper [16]. A huge amount of effort has gone into the classification of all finite distancetransitive graphs. The classification naturally breaks into two parts, primitive and imprimitive. The main part of the problem is the classification of all finite distancetransitive graphs with primitive automorphism group, and it appears that this classification is nearly finished. For the imprimitive case, Smith [24] showed that the possibilities for nontrivial blocks of imprimitivity are severely limited and that a given imprimitive distance-transitive graph can in a sense be reduced to a primitive distance-transitive graph. Van Bon and Brouwer [5] and Hemmeter [14; 15] carried through the reverse of Smith’s theorem, classifying for most of the known primitive distance-transitive graphs any associated imprimitive distance-transitive graphs they might have. In [3] the present authors gave a precise version of Smith’s theorem which implies that any unknown imprimitive distance-transitive graph must arise from a primitive distance-transitive graph of diameter at least 2 and valency at least 3. In the present paper, we return to the work of van Bon and Brouwer [5] and Hemmeter [14; 15] and show that, starting from each of the known distance-transitive graphs of diameter and valency at least 3, there are no surprises—the only associated imprimitive graphs are ones already known and in the literature (see [7]). The terminology and results will be made precise in the next section. In particular we give a precise version of Smith’s theorem (following [3]) and describe how the present results fit into the general problem of classifying all distance-transitive graphs. In Section 3 we give various general results about the parameters of a distance-regular graph, particularly those that are imprimitive in one of the two ways specified by Smith’s theorem. Section 4 discusses various of the geometries, designs, and codes often used in constructing and describing the graphs under consideration. Of particular import are the gamma spaces introduced by Higman.