Dual Bipartite Q-Polynomial Distance-Regular Graphs and Dual Uniform Structures - Monzillo, Giusy (The Electronic Journal of Combinatorics, 2024)
Let Γ denote a dual bipartite Q-polynomial distance-regular graph with vertex set X and diameter D ≥ 3. Fix x 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 Γ 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.