The asymptotic value of Randic index for trees

TL;DR

This paper determines the asymptotic value of the Randić index for large trees and confirms Fajtlowicz's conjecture for almost all trees and connected graphs using probabilistic methods.

Contribution

It establishes the asymptotic behavior of the Randić index for trees and proves Fajtlowicz's conjecture holds for almost all trees and connected graphs.

Findings

The distribution of double-star occurrences in random trees is normal.

The asymptotic value of the Randić index for trees is derived.

Fajtlowicz's conjecture is validated for almost all trees and connected graphs.

Abstract

Let $\mathcal{T}_n$ denote the set of all unrooted and unlabeled trees with $n$ vertices, and $(i,j)$ a double-star. By assuming that every tree of $\mathcal{T}_n$ is equally likely, we show that the limiting distribution of the number of occurrences of the double-star $(i,j)$ in $\mathcal{T}_n$ is normal. Based on this result, we obtain the asymptotic value of Randi\'c index for trees. Fajtlowicz conjectured that for any connected graph the Randi\'c index is at least the average distance. Using this asymptotic value, we show that this conjecture is true not only for almost all connected graphs but also for almost all trees.