A characterization of the Odd graphs and the doubled Odd graphs with a few of their intersection numbers

We show several inequalities for intersection numbers of distance-regular graphs. As an application of them we characterize the Odd graphs and the doubled Odd graphs with a few of their intersection numbers. In particular, we prove that the diameter d of a bipartite distance-regular graph of valency k and girth 2 r + 2 ≥ 6 is bounded by d ≤ [ k + 2 2 ] r + 1 if it is not the doubled Odd graph.