Bipartite Q-polynomial distance-regular graphs and uniform posets

2011/08/11 by Miklavic, Stefko, Terwilliger, Paul · 1 citation
#05C50 #05E30 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1108.2484

Abstract

Let \G 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 \G 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 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.

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• Raising and lowering maps for tridiagonal pairs

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