Dual Bipartite $Q$-Polynomial Distance-Regular Graphs and Dual Uniform Structures

Let $\Gamma$ denote a dual bipartite $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D \geq 3$. Fix $x \in X$, and let $L^*$ and $R^*$ denote the corresponding dual lowering and dual raising matrix, respectively. We show that a certain linear dependency among $R^* L^{* 2}, L^* R^* L^*, L^{* 2} R^*, L^*$ holds, and determine whether this linear dependency endow $\Gamma$ with a dual uniform or dual strongly uniform structure. Precisely, except for two special cases a dual uniform structure is always attained, and except for four special cases a dual strongly uniform structure is always attained.