A characterization of some distance-regular graphs by strongly closed subgraphs

In this paper we study a distance-regular graph ΓΓ of diameter d≥4d≥4 such that for any given pair of vertices at distance d−1d−1 there exists a strongly closed subgraph of diameter d−1d−1 containing them. We prove several inequalities for intersection numbers of ΓΓ. We show that if the equalities hold in some of these inequalities, then ΓΓ is either the Odd graph, the doubled Odd graph, the doubled Grassmann graph, the Hamming graph or the dual polar graph.