The last subconstituent of a bipartite Q-polynomial distance-regular graph

Let Γ denote a bipartite distance-regular graph with diameter D≥3. Fix any vertex x and let ΓD=ΓD(x) denote the set of vertices at distance D from x. Let ΓD2=ΓD2(x) denote the graph with vertex set ΓD, and edge set consisting of all pairs of vertices in ΓD which are at distance 2 in Γ. In this paper, we assume Γ is Q-polynomial and show ΓD2 is distance-regular and Q-polynomial. We compute the intersection numbers of ΓD2 from the intersection numbers of Γ. To obtain our results, we use a characterization of the Q-polynomial property due to Terwilliger.