New results on the Wiener index of trees with a given diameter

TL;DR

This paper characterizes the trees with fixed diameter and order that maximize the Wiener index, showing that balanced double brooms are optimal under certain size constraints.

Contribution

It proves that balanced double brooms uniquely maximize the Wiener index among trees with fixed diameter and size within a specific range.

Findings

Balanced double brooms maximize the Wiener index for certain tree sizes.

The results are sharp up to a small constant.

Provides a precise characterization of extremal trees for the Wiener index.

Abstract

We study the Wiener index of a class of trees with fixed diameter and order. A double broom is a tree such that there exist two vertices $u$ and $v$, such that each leaf of $T$ is adjacent to $u$ or $v$. We prove that for a tree $T$ of diameter $d$ and (sufficiently large) order $n$ such that $n\leq d-2+4 \left\lfloor \sqrt{ \frac{d-1}{2}} \right\rfloor$, $T$ has maximum Wiener index (in the class of trees of diameter $d$ and order $n$) if and only if $T$ is a balanced double broom. Our results are sharp up to a small constant.