Bipartite Q-polynomial distance-regular graphs and uniform posets

@article{Miklavic2011BipartiteQD,
  title={Bipartite Q-polynomial distance-regular graphs and uniform posets},
  author={Stefko Miklavic and Paul M. Terwilliger},
  journal={Journal of Algebraic Combinatorics},
  year={2011},
  volume={38},
  pages={225-242}
}
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 RL2, LRL, L2R, 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 a uniform or strongly… 

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