Recursions Satisfied by Families of Determinants with Applications to Resistance Distance

TL;DR

This paper introduces a semi-automatic algorithm to derive recursions for determinants related to resistance distances in graphs, enabling explicit formulas and proving conjectures for complex graph families.

Contribution

It presents a novel six-step algorithm for systematically deriving recursions for determinants, facilitating resistance distance calculations in complex graph families.

Findings

Derived recursions for resistance distances in complex graphs

Proved the 1/14 asymptotic conjecture for linear 3--trees

Provided a survey connecting Fibonacci numbers with resistance distances

Abstract

The main contribution of this paper is a six-step semi-automatic algorithm that obtains a recursion satisfied by a family of determinants by systematically and iteratively applying Laplace expansion to the underlying matrix family. The recursion allows explicit computation of the Binet form providing a closed formula for resistance distance between two specified nodes in a family of graphs. This approach is particularly suited for graph families with complex structures; the method is used to prove the 1 over 14 conjectured asymptotic formula for linear 3--trees. Additionally, although the literature on recursive formulas for resistance distances is quite large, the Fibonacci Quarterly and the Proceedings have almost no such results despite the fact that many recursions related to resistance distances involve the Fibonacci numbers. Therefore, a secondary purpose of the paper is to provide a brief introductory survey of graph families, accompanied by figures, Laplacian matrices, and typical recursive results, supported by a modest bibliography of current papers on many relevant graph families, in the hope to involve Fibonaccians in this active and beautiful field.