On the $Q$-polynomial property of bipartite graphs admitting a uniform structure
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| Format: | Preprint |
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2025
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| _version_ | 1866910857628221440 |
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| author | Fernández, Blas Maleki, Roghayeh Miklavič, Štefko Monzillo, Giusy |
| author_facet | Fernández, Blas Maleki, Roghayeh Miklavič, Štefko Monzillo, Giusy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02339 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $Q$-polynomial property of bipartite graphs admitting a uniform structure Fernández, Blas Maleki, Roghayeh Miklavič, Štefko Monzillo, Giusy Combinatorics 05E99, 05C50 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. |
| title | On the $Q$-polynomial property of bipartite graphs admitting a uniform structure |
| topic | Combinatorics 05E99, 05C50 |
| url | https://arxiv.org/abs/2503.02339 |