Ordering trees having small reverse Wiener indices

TL;DR

This paper investigates the reverse Wiener index of trees, identifying those with the second and third smallest indices among n-vertex trees and characterizing their structures for n≥6.

Contribution

It determines the second and third smallest reverse Wiener indices of n-vertex trees and characterizes the trees attaining these values.

Findings

Identified trees with second and third smallest reverse Wiener indices.

Characterized the structure of these trees for n≥6.

Confirmed the star as the unique tree with the smallest reverse Wiener index.

Abstract

The reverse Wiener index of a connected graph $G$ is a variation of the well-known Wiener index $W(G)$ defined as the sum of distances between all unordered pairs of vertices of $G$. It is defined as $\Lambda(G)=\frac{1}{2}n(n-1)d-W(G)$, where $n$ is the number of vertices, and $d$ is the diameter of $G$. We now determine the second and the third smallest reverse Wiener indices of $n$-vertex trees and characterize the trees whose reverse Wiener indices attain these values for $n\ge 6$ (it has been known that the star is the unique tree with the smallest reverse Wiener index).