Bipartite Q-Polynomial Distance-Regular Graphs

Let Γ denote a bipartite distance-regular graph with diameter D≥12. We show Γ is Q-polynomial if and only if one of the following (i)–(iv) holds: (i) Γ is the ordinary 2D-cycle. (ii) Γ is the Hamming cube H(D,2). (iii) Γ is the antipodal quotient of H(2D,2). (iv) The intersection numbers of Γ satisfy [inline-graphic not available: see fulltext] where q is an integer at least 2. We obtain the above result using the Terwilliger algebra of Γ.