Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree
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| Format: | Preprint |
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2025
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| author | Evans, Emily J. Hendel, Russell Jay |
| author_facet | Evans, Emily J. Hendel, Russell Jay |
| contents | Barret, Evans, and Francis conjectured that if $G$ is the straight linear 3-tree with $n$ vertices and $H$ is the straight linear 3-tree with $n+1$ vertices then
\[\lim_{n\rightarrow \infty} r_{H} (1, n+1) - r_G(1,n) = \frac{1}{14},\] where $r_G(u,v)$ and $r_H(u,v)$ are the resistance distance between vertices $u$ and $v$ in graphs $G$ and $H$ respectively. In this paper, we prove the conjecture by looking at the determinants of deleted Laplacian matrices. The proof uses a Laplace expansion method on a family of determinants to determine the underlying recursion this family satisfies and then uses routine linear algebra methods to obtain an exact Binet formula for the $n$-th term. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_20539 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree Evans, Emily J. Hendel, Russell Jay Combinatorics 11B37 11B39 94C15 Barret, Evans, and Francis conjectured that if $G$ is the straight linear 3-tree with $n$ vertices and $H$ is the straight linear 3-tree with $n+1$ vertices then \[\lim_{n\rightarrow \infty} r_{H} (1, n+1) - r_G(1,n) = \frac{1}{14},\] where $r_G(u,v)$ and $r_H(u,v)$ are the resistance distance between vertices $u$ and $v$ in graphs $G$ and $H$ respectively. In this paper, we prove the conjecture by looking at the determinants of deleted Laplacian matrices. The proof uses a Laplace expansion method on a family of determinants to determine the underlying recursion this family satisfies and then uses routine linear algebra methods to obtain an exact Binet formula for the $n$-th term. |
| title | Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree |
| topic | Combinatorics 11B37 11B39 94C15 |
| url | https://arxiv.org/abs/2505.20539 |