Bipartite Distance-Transitive Doubles with Primitive Halved of Diameter Two

Abstract

As a consequence of a famous theorem by Derek Smith, an unknown distance-transitive graph is either primitive of diameter at least two and valency at least three or is antipodal, bipartite, or both. In the imprimitive cases an unknown graph must have a primitive core of diameter at least two and valency at least three. It seems that the known list of primitive graphs is complete. Here, starting from earlier work by Hemmeter we find every bipartite distance-transitive double whose primitive halved is one of the known distance-transitive graphs of diameter two and valency at least three.