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

AbstractWe 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 2r+2≥6 is bounded by d≤[k+22]r+1 if it is not the doubled Odd graph