The distance-regular graphs of valency four

In this note we report on a computer search that proves that each distance-regular graph of valency four has known parameters. Here we describe first the known examples, next how putative arrays were disposed of, and finally how the search could be limited to a manageable number of arrays. The distance-regular graphs of valency 3 have been determined by Biggs et al. [6]. Bannai and Ito worked on the general project of bounding the diameter of a distance-regular graph as a function of its valency k. They succeeded in the bipartite case [3] and in case k = 4[4](1). This means that finding the feasible arrays for distance-regular graphs of valency 4 was reduced to a finite amount of work, but the diameter bounds obtained were not small enough to straightforwardly settle this case. In this note we obtain some additional conditions, and thus reduce the parameter space to be searched, and describe a way to test a parameter set using (small) integer arithmetic, thus avoiding accuracy problems. Our notation for distance-regular graphs is standard (cf. [1, 5, 8]).