Quadratic form estimations for Hessian matrices of resistance distance and Kirchhoff index of positive-weighted graphs
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911488283770880 |
|---|---|
| author | Li, Yu Sun, Lizhu Bu, Changjiang |
| author_facet | Li, Yu Sun, Lizhu Bu, Changjiang |
| contents | Let $G^{w}=(V,E,w)$ be a positive-weighted graph with the weight $w(e)>0$ for all $e\in E$. The weighted graph $G^{\widetilde{w}}=(V,E,\widetilde{w})$ is called a hyper-dual number weighted graph, where the weight $\widetilde{w}(e)=w(e)+Δw(e)(\varepsilon+\varepsilon^{*})$ is a hyper dual number, $Δw(e)$ is a real number, $\varepsilon$ and $\varepsilon^{*}$ are two dual units, $e\in E$. In this paper, we give a representation for the Moore-Penrose inverse of the Laplacian matrix, and calculation formulas for the resistance distance and Kirchhoff index of $G^{\widetilde{w}}$, respectively. We establish quadratic forms of the Hessian matrices for the resistance distance and Kirchhoff index of $G^{w}$ via generalized matrix inverses. We further derive explicit bounds on the eigenvalues of the Hessian matrices for the resistance distance and the Kirchhoff index of $G^{w}$ in terms of graph parameters. We also prove that the Kirchhoff index of a positive-weighted graph with bounded edge weights is strongly convex on its edge weight vector. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04963 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quadratic form estimations for Hessian matrices of resistance distance and Kirchhoff index of positive-weighted graphs Li, Yu Sun, Lizhu Bu, Changjiang Combinatorics 05C09, 05C12, 05C50, 15A10 Let $G^{w}=(V,E,w)$ be a positive-weighted graph with the weight $w(e)>0$ for all $e\in E$. The weighted graph $G^{\widetilde{w}}=(V,E,\widetilde{w})$ is called a hyper-dual number weighted graph, where the weight $\widetilde{w}(e)=w(e)+Δw(e)(\varepsilon+\varepsilon^{*})$ is a hyper dual number, $Δw(e)$ is a real number, $\varepsilon$ and $\varepsilon^{*}$ are two dual units, $e\in E$. In this paper, we give a representation for the Moore-Penrose inverse of the Laplacian matrix, and calculation formulas for the resistance distance and Kirchhoff index of $G^{\widetilde{w}}$, respectively. We establish quadratic forms of the Hessian matrices for the resistance distance and Kirchhoff index of $G^{w}$ via generalized matrix inverses. We further derive explicit bounds on the eigenvalues of the Hessian matrices for the resistance distance and the Kirchhoff index of $G^{w}$ in terms of graph parameters. We also prove that the Kirchhoff index of a positive-weighted graph with bounded edge weights is strongly convex on its edge weight vector. |
| title | Quadratic form estimations for Hessian matrices of resistance distance and Kirchhoff index of positive-weighted graphs |
| topic | Combinatorics 05C09, 05C12, 05C50, 15A10 |
| url | https://arxiv.org/abs/2603.04963 |