Harmonic-Arithmetic Index of (Molecular) Trees

TL;DR

This paper introduces the harmonic-arithmetic (HA) index for molecular trees, characterizes extremal graphs within this class, and explores its mathematical and application significance in graph invariants.

Contribution

It defines the HA index based on harmonic and arithmetic means and characterizes graphs with maximum or minimum HA index among molecular trees.

Findings

Graphs with extremal HA index are fully characterized.

The HA index unifies and extends known graph invariants.

Results have implications for molecular structure analysis.

Abstract

Let $G$ be a graph. Denote by $d_x$, $E(G)$, and $D(G)$ the degree of a vertex $x$ in $G$, the set of edges of $G$, and the degree set of $G$, respectively. This paper proposes to investigate (both from mathematical and applications points of view) those graph invariants of the form $\sum_{uv\in E(G)}\varphi(d_v,d_w)$ in which $\varphi$ can be defined either using well-known means of $d_v$ and $d_w$ (for example: arithmetic, geometric, harmonic, quadratic, and cubic means) or by applying a basic arithmetic operation (addition, subtraction, multiplication, and division) on any of two such means, provided that $\varphi$ is a non-negative and symmetric function defined on the Cartesian square of $D(G)$. Many existing well-known graph invariants can be defined in this way; however, there are many exceptions too. One of such uninvestigated graph invariants is the harmonic-arithmetic (HA) index, which is obtained from the aforementioned setting by taking $\varphi$ as the ratio of the harmonic and arithmetic means of $d_v$ and $d_w$. A molecular tree is a tree whose maximum degree does not exceed four. Given the class of all (molecular) trees with a fixed order, graphs that have the largest or least value of the HA index are completely characterized in this paper.