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

摘要

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