On the Q-polynomial property of bipartite graphs admitting a uniform structure

Let denote a finite, connected graph with vertex set X. Fix x X and let 3 denote the eccentricity of x. For mutually distinct scalars \^*ᵢ\₈=₀^ define a diagonal matrix A^*=A^* (^*₀, ^*₁, , ^*_) MX (R) as follows: for y X we let (A^*) ₘₘ = ^* (ₗ, ₘ), where denotes the shortest path length distance function of. We say that A^* is a dual adjacency matrix candidate of with respect to x if the adjacency matrix A MX (R) of and A^* satisfy A³ A^* - A^* A³+ (+1) (A A^* A² - A² A^* A) = (A²A^*-A^*A²) + (A A^* - A^* A) for some scalars, , R. Assume now that is uniform with respect to x in the sense of Terwilliger Coding theory and design theory, Part I, IMA Vol. Math. Appl. , 20, 193-212 (1990). In this paper, we give sufficient conditions on the uniform structure of, such that admits a dual adjacency matrix candidate with respect to x. As an application of our results, we show that the full bipartite graphs of dual polar graphs are Q-polynomial.