Resistance distance in straight linear 2-trees

TL;DR

This paper derives explicit formulas for resistance distances in straight linear 2-trees, revealing their properties and potential for link prediction, and introduces new analytical techniques involving Fibonacci and Lucas numbers.

Contribution

It provides the first explicit resistance distance formulas for a nontrivial family of graphs with unbounded diameter, using Fibonacci and Lucas number identities.

Findings

Resistance distance between vertices 1 and n tends to infinity as n increases.

Maximum resistance distance occurs between vertices 1 and n.

Resistance distance ordering aligns with intuitive graph distances.

Abstract

We consider the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$. We call $G_n$ the straight linear 2-tree on $n$ vertices. Using $\Delta$--Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance $r_{G_n}(i,j)$ between any two vertices $i$ and $j$ of $G_n$. To our knowledge $\{G_n\}_{n=3}^\infty$ is the first nontrivial family with diameter going to $\infty$ for which all resistance distances have been explicitly calculated. Our result also gives formulae for the number of spanning trees and 2-forests in a straight linear 2-tree. We show that the maximal resistance distance in $G_n$ occurs between vertices 1 and $n$ and the minimal resistance distance occurs between vertices $n/2$ and $n/2+1$ for $n$ even (with a similar result for $n$ odd). It follows that $r_n(1,n) \to \infty$ as $n \to \infty$. Moreover, our explicit formula makes it possible to order the non-edges of $G_n$ exactly according to resistance distance, and this ordering agrees with the intuitive notion of distance on a graph. Consequently, $G_n$ is a geometric graph with entirely different properties than the random geometric graphs investigated in [6]. These results for straight linear 2-trees along with an example of a bent linear 2-tree and empirical results for additional graph classes convincingly demonstrate that resistance distance should not be discounted as a viable method for link prediction in geometric graphs.