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

Abstract

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