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

Let $\Gamma$ denote a finite, connected graph with vertex set $X$. Fix $x \in X$ and let $\varepsilon \ge 3$ denote the eccentricity of $x$. For mutually distinct scalars $\{\theta^*_i\}_{i=0}^\varepsilon$ define a diagonal matrix $A^*=A^*(\theta^*_0, \theta^*_1, \ldots, \theta^*_{\varepsilon}) \in M_X(\mathbb{R})$ as follows: for $y \in X$ we let $(A^*)_{yy} = \theta^*_{\partial(x,y)}$, where $\partial$ denotes the shortest path length distance function of $\Gamma$. We say that $A^*$ is a dual adjacency matrix candidate of $\Gamma$ with respect to $x$ if the adjacency matrix $A \in M_X(\mathbb{R})$ of $\Gamma$ and $A^*$ satisfy $$A^3 A^* - A^* A^3+(\beta+1)( A A^* A^2 - A^2 A^* A)= \gamma(A^2A^*-A^*A^2)+\rho( A A^* - A^* A)$$ for some scalars $\beta, \gamma, \rho\in \mathbb{R}$. Assume now that $\Gamma$ 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 $\Gamma$, such that $\Gamma$ 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.