Geodesic
Bipartite $Q$-polynomial distance-regular graphs and uniform posets
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 225-242.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $\Gamma $ denote a bipartite distance-regular graph with vertex set X and diameter D$\geq 3$. Fix x$\in X$ and let L (resp., R) denote the corresponding lowering (resp., raising) matrix. We show that each Q-polynomial structure for $\Gamma $ yields a certain linear dependency among RL $^{2}, LRL,$ L $^{2}$ R, L. Define a partial order $\leq $on X as follows. For y,z$\in X$ let y$\leq z$ whenever $\partial $(x,y)+$\partial $(y,z)=$\partial $(x,z), where $\partial $ 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.
Keywords: distance-regular graphs, Q-polynomial structure, uniform posets 
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     author = {Miklavi\v{c}, \v{S}tefko and Terwilliger, Paul},
     title = {Bipartite $Q$-polynomial distance-regular graphs and uniform posets},
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Miklavič, Štefko; Terwilliger, Paul. Bipartite $Q$-polynomial distance-regular graphs and uniform posets. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 225-242. http://geodesic.mathdoc.fr/item/JAC_2013__38_2_a11/