Bounding the diameter of a distance regular graph by a function of \(k_ d\). II
[For Part I see the author, Graphs Comb. 7, No. 4, 363-375 (1991; Zbl 0755.05057).] Let \(G\) be a distance regular graph of diameter \(d\) and let \(u\), \(v\) be vertices at distance \(d\) in \(G\). Let \(a_ p\) be the number of neighbours of \(u\) which have distance \(d\) from \(v\), and let \(k_ p\) be the total number of vertices at distance \(d\) from \(v\). It is shown that if \(a_ d \neq 0\) and the valency of \(G\) is greater than 2, then the diameter of \(G\) is bounded by a function depending only on \(k_ d\). In particular, for a fixed \(k_ d\) there are only finitely many distance regular graphs of valency \(>2\) with \(a_ d \neq 0\).
scientific article; zbMATH DE number 701273