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

2025/05/26 by Evans, Emily J., Hendel, Russell Jay
#11B37 11B39 94C15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.20539

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→ ∞} r_H (1, n+1) - r_G(1,n) = (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.

Citations

• Recursions Satisfied by Families of Determinants with Applications to Resistance Distance
• Algorithmic techniques for finding resistance distances on structured graphs
• Resistance distance, Kirchhoff index, and Kemeny's constant in flower graphs
• Resistance distance in straight linear 2-trees
• Graph Sparsification by Effective Resistances

Related

• Recursions Satisfied by Families of Determinants with Applications to Resistance Distance
• Resistance distance in straight linear 2-trees
• Resistance distance in bent linear 2-trees
• Solution to a conjecture on resistance distances of block tower graphs
• Homogeneous linear recurrence relations of the determinants of distance matrices of trees