On the transmission irregular trees with the maximum Wiener index

TL;DR

This paper investigates the structure of transmission irregular trees that maximize the Wiener index, providing solutions for all odd and most even tree sizes, with unique extremal solutions identified as chemical trees.

Contribution

It solves the Wiener index maximization problem for transmission irregular trees of any given order, identifying unique extremal chemical trees for almost all sizes.

Findings

Maximized Wiener index for all odd tree sizes.

Almost all even tree sizes have a unique solution.

Extremal trees are chemical trees.

Abstract

The transmission of a vertex $v$ in a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is transmission irregular. The Wiener index $W(G)$ of a graph $G$ is the sum of all distances between all unordered pairs of vertices in $G$, which has another formula as the half of the sum of transmissions of all vertices of $G$. In this paper, we consider the Wiener index maximization problem on the set of transmission irregular trees of a given order $n \in \mathbb{N}$. We solve the problem for all odd values of $n$ and for almost all even values of $n$. Each resolved extremal problem has a unique solution that is a chemical tree.