On the $Q$-polynomial property of bipartite graphs admitting a uniform structure
Blas Fernández, Roghayeh Maleki, Štefko Miklavič, Giusy Monzillo
TL;DR
The paper develops a framework to certify Q-polynomial structure in graphs with a uniform bipartite framework by introducing dual adjacency matrix candidates $A^*(\theta^*_0,\dots,\theta^*_\varepsilon)$ and a tridiagonal relation linking $A$ and $A^*$. It leverages Terwilliger algebras, the lowering/raising operators, and a uniform structure $(U,f)$ to derive sufficient conditions under which such an $A^*$ exists, along with explicit parameter relations $\beta,\gamma,\rho$. As a key application, it proves that the full bipartite graphs of dual polar graphs are $Q$-polynomial, providing explicit eigenvalues and dual eigenstructure. The results extend Terwilliger's approach to broader classes of graphs beyond distance-regular ones and offer a concrete method to certify the $Q$-polynomial property for bipartite, uniformly structured graphs.
Abstract
Let $Γ$ 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 $\{θ^*_i\}_{i=0}^\varepsilon$ define a diagonal matrix $A^*=A^*(θ^*_0, θ^*_1, \ldots, θ^*_{\varepsilon}) \in M_X(\mathbb{R})$ as follows: for $y \in X$ we let $(A^*)_{yy} = θ^*_{\partial(x,y)}$, where $\partial$ 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 $A \in M_X(\mathbb{R})$ of $Γ$ and $A^*$ satisfy $$ A^3 A^* - A^* A^3+(β+1)( A A^* A^2 - A^2 A^* A)= γ(A^2A^*-A^*A^2)+ρ( A A^* - A^* A) $$ for some scalars $β, γ, ρ\in \mathbb{R}$. 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.