On distance-regular graphs with fixed valency. III
This paper is a sequel to parts I and II in Graphs Comb. 3, 95-109 (1987; Zbl 0626.05050), and ibid. 4, No.3, 219-228 (1988), respectively, but will be read mostly independent of them as long as the logical dependencies are concerned. The ultimate purpose of this series of papers is to obtain the following two assertions: (i) to bound the diameter d of any distance-regular graph by a certain function of valency k (k\(\geq 3)\), (ii) to complete the classification of distance-regular graphs with small k. (It is expected that (ii) follows generally from (i).) In this paper we answer assertion (i) completely for the bipartite case. (The graph \(\Gamma\) is said to be bipartite if the vertex set X is divided into two non-empty subsets \(X_ 1\) and \(X_ 2\) with no edges inside \(X_ i\) \((i=1,2)\). In the distance-regular case, \(\Gamma\) is bipartite if and only if \(a_ i=0\) for all i.) The general case (i.e., the nonbipartite case) will be treated in the subsequent papers of this sequel.
scientific article; zbMATH DE number 4108822