Bipartite Q-polynomial distance-regular graphs and uniform posets

Štefko Miklavič, Paul Terwilliger
2012 Journal of Algebraic Combinatorics  
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 a
more » ... rm 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.
doi:10.1007/s10801-012-0401-1 fatcat:dwea5vh66bgm3n6hw4xb3acwki