The distance-regular graphs of valency four
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 \textit{N. L. Biggs}, \textit{A. G. Boshier}, and \textit{J. Shawe-Taylor} [J. Lond. Math. Soc., II. Ser. 33, 385-394 (1986; Zbl 0576.05039)]. \textit{E. Bannai} and \textit{T. Itô} 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 [J. Algebra 107, 43-52 (1987; Zbl 0677.05060)] and in case \(k= 4\) [see Eur. J. Comb. 10, No. 2, 137-148 (1989; Zbl 0677.05061)]. 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.
scientific article; zbMATH DE number 1319550