, Volume 38, Issue 2, pp 225-242
Date: 09 Oct 2012

Bipartite Q-polynomial distance-regular graphs and uniform posets

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Abstract

Let Γ denote a bipartite distance-regular graph with vertex set X and diameter D≥3. Fix xX 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,zX let yz 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.