On Bipartite Q-Polynomial Distance-Regular Graphs with Diameter 9, 10, or 11

Let Γ denote a bipartite distance-regular graph with diameter D. In [J. S. Caughman, Bipartite Q-polynomial distance-regular graphs, Graphs Combin. 20 (2004), 47–57], Caughman showed that if D > 12, then 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 the Hamming cube H(2D, 2), (iv) the intersection numbers of Γ satisfy ci = (q i − 1)/(q− 1), bi = (qD − qi)/(q − 1) (0 6 i 6 D), where q is an integer at least 2. In this paper we show that the above result is true also for bipartite distance-regular graphs with D ∈ {9, 10, 11}.