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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2403.06096 |
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| _version_ | 1866914709196767232 |
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| author | Xu, Si-Ao Zhou, Huan Pan, Xiang-Feng |
| author_facet | Xu, Si-Ao Zhou, Huan Pan, Xiang-Feng |
| contents | Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The resistance distance $R_G(x,y)$ between two vertices $x,y$ of $G$ is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of $G$ is replaced by a unit resistor. The resistance spectrum $\mathrm{RS}(G)$ of a graph $G$ is the multiset of the resistance distances of all pairs of vertices in the graph. This paper presents a method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer $k$, there exist at least $2^k$ graphs with the same resistance spectrum. Furthermore, it is shown that for $n \geq 10$, there are at least $2(n-9) p(n-9)$ pairs of graphs of order $n$ with the same resistance spectrum, where $p(n-9)$ is the number of partitions of the integer $n-9$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06096 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A method for constructing graphs with the same resistance spectrum Xu, Si-Ao Zhou, Huan Pan, Xiang-Feng Combinatorics 05C12 Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The resistance distance $R_G(x,y)$ between two vertices $x,y$ of $G$ is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of $G$ is replaced by a unit resistor. The resistance spectrum $\mathrm{RS}(G)$ of a graph $G$ is the multiset of the resistance distances of all pairs of vertices in the graph. This paper presents a method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer $k$, there exist at least $2^k$ graphs with the same resistance spectrum. Furthermore, it is shown that for $n \geq 10$, there are at least $2(n-9) p(n-9)$ pairs of graphs of order $n$ with the same resistance spectrum, where $p(n-9)$ is the number of partitions of the integer $n-9$. |
| title | A method for constructing graphs with the same resistance spectrum |
| topic | Combinatorics 05C12 |
| url | https://arxiv.org/abs/2403.06096 |