On bipartite distance-regular graphs with a strongly closed subgraph of diameter three

Let Γ denote a distance-regular graph with a strongly closed regular subgraph Y. Hosoya and Suzuki R. Hosoya, H. Suzuki, Tight distance-regular graphs with respect to subsets, European J. Combin. 28 (2007) 61-74] showed an inequality for the second largest and least eigenvalues of Γ in the case Y is of diameter 2. In this paper, we study the case when Γ is bipartite and Y is of diameter 3, and obtain an inequality for the second largest eigenvalue of Γ. Moreover, we characterize the distance-regular graphs with a completely regular strongly closed subgraph H(3,2).