Exponential Harmonic Index and Its Applications in Structure Property Modeling

Topological indices, invariant under symmetry transformations that preserve a graph's connectivity, are fundamental tools in mathematical chemistry. By capturing intrinsic symmetries and connectivity patterns, these indices provide insightful analyses of molecular stability, reactivity, and other fundamental properties, making them indispensable in cheminformatics and theoretical chemistry. Among these, the harmonic index ($H$) is important in both chemistry and mathematics. It is a modification of the Randić index, widely recognized as a highly effective invariant in investigations of structure–property relationships. The $H$ index of a graph $G$ is formulated as

$H=H\left(G\right)=\sum _{v_i{v}_j\in E\left(G\right)}\frac{2}{d_i+d_j}$

where $d_j$ denotes the degree of the vertex $v_j$. In recent years, various exponential vertex-degree-based topological indices have been reported. In this paper, we define the exponential harmonic index ($EH$) as follows:

$EH=EH\left(G\right)=\sum _{v_i{v}_j\in E\left(G\right)}{e}^{\frac{2}{d_i+d_j}}$

The exponential harmonic index ($EH$) is investigated here from both chemical and mathematical perspectives. We examine the $EH$ index's ability to predict various physicochemical properties through quantitative structure-property relationship (QSPR) analysis. Additionally, we describe the extremal trees with respect to $EH$. Furthermore, the maximal tree for $EH$ is characterized in relation to a given maximum degree.