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

Let ΓΓ denote a distance-regular graph with a strongly closed regular subgraph YY. 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 ΓΓ in the case YY is of diameter 2. In this paper, we study the case when ΓΓ is bipartite and YY is of diameter 3, and obtain an inequality for the second largest eigenvalue of ΓΓ. Moreover, we characterize the distance-regular graphs with a completely regular strongly closed subgraph H(3,2)H(3,2).