On bipartite Q -polynomial distance-regular graphs with c2=1c2=1

Let ΓΓ denote a bipartite Q  -polynomial distance-regular graph with diameter d⩾3d⩾3, valency k⩾3k⩾3 and intersection number c2=1c2=1. We show that ΓΓ has a certain equitable partition of its vertex set which involves 4d-44d-4 cells. We use this partition to show that the intersection numbers of ΓΓ satisfy the following divisibility conditions:ci+1-1dividesci(ci-1)for2⩽i⩽d-1,bi-1-1dividesbi(bi-1)for1⩽i⩽d-1.Using these divisibility conditions we show ΓΓ does not exist if d=4d=4.