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 ) .