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 dd of a bipartite distance-regular graph of valency kk and girth 2r+2≥62r+2≥6 is bounded by d≤[k+22]r+1 if it is not the doubled Odd graph.