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

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 $$\underset{n\to \mathrm{\infty }}{\lim }{r}_H(1,n+1)-r_G(1,n)=\frac{1}{14},$$ 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.