Bounding the diameter of a distance regular graph by a function of \(k_ d\) (Q1180676)

For a distance regular graph \(G\) of diameter \(d\) let \(G_ d(u)\) be the subgraph of \(G\) induced by the set of all vertices at distance \(d\) from \(u\). \(G\) is said to be thin if \(G_ d(u)\) is a union of cliques for every vertex \(u\) of \(G\). The structure of thin distance regular graphs \(G\) is investigated in which all cliques in \(G_ d(u)\) are non-trivial. For those distance regular graphs \(G\) that are not thin it is shown that their diameter is bounded above by a function depending only on the number of vertices in \(G_ d(u)\).