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 \[limn→ ∞ rH (1, n+1) - r_G(1,n) = 1/14,\] where rG(u,v) and rH(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.