The resistance distance of a dual number weighted graph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909502139269120 |
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| author | Li, Yu Sun, Lizhu Bu, Changjiang |
| author_facet | Li, Yu Sun, Lizhu Bu, Changjiang |
| contents | For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the perturbation of the unit resistor on edge $e$ of $G$. For a connected dual number weighted graph $G^{w}$, we give some expressions and block representations of generalized inverses of the Laplacian matrix of $G^{w}$. And using these results, we derive the explicit formulas of the resistance distance and Kirchhoff index of $G^{w}$. We give the perturbation bounds for the resistance distance and Kirchhoff index of $G$. In particular, when only the edge $e=\{i,j\}$ of $G$ is perturbed, we give the perturbation bounds for the Kirchhoff index and resistance distance between vertices $i$ and $j$ of $G$, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_13455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The resistance distance of a dual number weighted graph Li, Yu Sun, Lizhu Bu, Changjiang Combinatorics 05C09, 05C12, 05C50, 15A10 For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the perturbation of the unit resistor on edge $e$ of $G$. For a connected dual number weighted graph $G^{w}$, we give some expressions and block representations of generalized inverses of the Laplacian matrix of $G^{w}$. And using these results, we derive the explicit formulas of the resistance distance and Kirchhoff index of $G^{w}$. We give the perturbation bounds for the resistance distance and Kirchhoff index of $G$. In particular, when only the edge $e=\{i,j\}$ of $G$ is perturbed, we give the perturbation bounds for the Kirchhoff index and resistance distance between vertices $i$ and $j$ of $G$, respectively. |
| title | The resistance distance of a dual number weighted graph |
| topic | Combinatorics 05C09, 05C12, 05C50, 15A10 |
| url | https://arxiv.org/abs/2502.13455 |