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

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