Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree

Emily J. Evans; Russell Jay Hendel

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arXiv · 2505.20539

Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree

Emily J. Evans·Russell Jay Hendel

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Abstract

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.

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BibTeX RIS

Emily J. Evans, Russell Jay Hendel. 2025-05-26. Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree. https://arxiv.org/abs/2505.20539

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