Proof of a Conjecture on the Growth of the Maximal Resistance Distance in a Linear 3--Tree
Emily J. Evans, Russell Jay Hendel
TL;DR
The paper addresses the conjecture on the asymptotic growth of the maximal resistance distance in straight linear 3-trees by developing a determinant-based Laplace-expansion framework for deleted Laplacian matrices. It derives recurrences for determinant families, obtains a Binet-form closed expression, and yields an exact closed form for the resistance distance $R(n)$ as a ratio of exponential terms in roots $r_i$, establishing that $R(n+1)-R(n) \to 1/14$ with geometric convergence of the error terms. The dominant root governs the asymptotics, while secondary roots contribute to the error, which is controlled via stable “sister” sequences. The method not only proves the conjecture but also provides a potentially general tool for computing resistance distances in other nearly banded graph families.
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.