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

TL;DR

This paper proves a conjecture about the asymptotic growth of resistance distance in linear 3-trees by analyzing determinants of Laplacian matrices, confirming the limit difference of 1/14.

Contribution

It provides a rigorous proof of the conjecture using Laplace expansion and linear algebra techniques to analyze resistance distances in linear 3-trees.

Findings

Confirmed the limit difference of resistance distances as 1/14

Developed a recursive formula for determinants of Laplacian matrices

Derived an exact Binet formula for the resistance distance sequence

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.