The Harary index of trees

TL;DR

This paper studies the Harary index of trees, identifying extremal trees with maximal or minimal indices under various structural constraints, and explores their relationships with other graph invariants.

Contribution

It characterizes extremal trees for the Harary index based on multiple parameters and establishes their connection with Wiener index and other graph properties.

Findings

Trees with maximal Harary index have minimal Wiener index.

Extremal trees are characterized for various parameters like degree, matching number, and diameter.

The paper provides partial orderings of starlike trees based on the Harary index.

Abstract

The Harary index of a graph $G$ is recently introduced topological index, defined on the reverse distance matrix as $H(G)=\sum_{u,v \in V(G)}\frac{1}{d(u,v)}$, where $d(u,v)$ is the length of the shortest path between two distinct vertices $u$ and $v$. We present the partial ordering of starlike trees based on the Harary index and we describe the trees with the second maximal and the second minimal Harary index. In this paper, we investigate the Harary index of trees with $k$ pendent vertices and determine the extremal trees with maximal Harary index. We also characterize the extremal trees with maximal Harary index with respect to the number of vertices of degree two, matching number, independence number, radius and diameter. In addition, we characterize the extremal trees with minimal Harary index and given maximum degree. We concluded that in all presented classes, the trees with maximal Harary index are exactly those trees with the minimal Wiener index, and vice versa.