Bipartite Q-polynomial distance-regular graphs and uniform posets

Let I" denote a bipartite distance-regular graph with vertex set X and diameter Da parts per thousand yen3. Fix xaX and let L (resp., R) denote the corresponding lowering (resp., raising) matrix. We show that each Q-polynomial structure for I" yields a certain linear dependency among RL (2), LRL, L (2) R, L. Define a partial order a parts per thousand currency sign on X as follows. For y,zaX let ya parts per thousand currency signz whenever a,(x,y)+a,(y,z)=a,(x,z), where a, denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.