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

Let Γ denote a finite, connected graph with vertex set X. Fix x ∈ X and let ε ≥ 3 denote the eccentricity of x. For mutually distinct scalars \θ^*ᵢ\ᵢ₌₀^ε define a diagonal matrix A^*=A^*(θ^*₀, θ^*₁, …, θ^*ε) ∈ MX(R) as follows: for y ∈ X we let (A^*)yy = θ^*∂(x,y), where ∂ denotes the shortest path length distance function of Γ. We say that A^* is a dual adjacency matrix candidate of Γ with respect to x if the adjacency matrix A ∈ MX(R) of Γ and A^* satisfy A³ A^* - A^* A³+(β+1)( A A^* A² - A² A^* A)= γ(A²A^*-A^*A²)+ρ( A A^* - A^* A) for some scalars β, γ, ρ∈ R. Assume now that Γ 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 Γ, such that Γ 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.