Almost-bipartite Q-polynomial Distance-regular Graphs And Uniform Posets

Let Γ =(X, R) be one of the(2D + 1)-gon, the folded(2D + 1)-cube and the Odd graph on a set of cardinality 2D + 1 with diameter D ≥ 3. It is known that Γ is an almost-bipartite Q-polynomial distance-regular graph. We first characterize the Qpolynomial structures of the folded(2D + 1)-cube by using the theory of Leonard pairs.Next, fix x ∈ X and define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ?(x, y) + ?(y, z) = ?(x, z), where ? denotes the path-length distance function.Then the partition {Γi(x)}D i=0of X is a grading of the poset(X, ≤). Let R(resp. L)denote the corresponding raising matrix(resp. lowering matrix). Then we show that there exists a certain linear dependency among RL~2, LRL, L~2 R and L for each given Q-polynomial structure of Γ. We call this dependency an R/L dependency structure.Finally, we determine whether the above R/L dependency structure gives this poset a uniform structure or strongly uniform structure.