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Let Γ denote a bipartite distance-regular graph with vertex set X and diameter D ≥ 3. Fix x ∈ X and let L (resp., R) denote the corresponding lowering (resp., raising) matrix. We show that each Q-polynomial structure for Γ yields a certain linear dependency among RL 2 , LRL, L 2 R, L. Define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ∂(x, y) + ∂(y, z) = ∂(x, z), where ∂ denotes path-length distance. We determine whether the above linear dependency gives this poset adoi:10.1007/s10801-012-0401-1 fatcat:dwea5vh66bgm3n6hw4xb3acwki