Effective resistance in planar graphs and continued fractions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912393076932608 |
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| author | Chan, Swee Hong Kontorovich, Alex Pak, Igor |
| author_facet | Chan, Swee Hong Kontorovich, Alex Pak, Igor |
| contents | For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{τ(G/e)}{τ(G)}$, where $τ(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedláček's inverse problem for the number of spanning trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective resistance in planar graphs and continued fractions Chan, Swee Hong Kontorovich, Alex Pak, Igor Combinatorics Number Theory For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{τ(G/e)}{τ(G)}$, where $τ(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedláček's inverse problem for the number of spanning trees. |
| title | Effective resistance in planar graphs and continued fractions |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2505.19168 |