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 ((Formula presented.)) 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 (Formula presented.) index of a graph (Formula presented.) is formulated as (Formula presented.) where (Formula presented.) denotes the degree of the vertex (Formula presented.). In recent years, various exponential vertex-degree-based topological indices have been reported. In this paper, we define the exponential harmonic index ((Formula presented.)) as follows: (Formula presented.) The exponential harmonic index ((Formula presented.)) is investigated here from both chemical and mathematical perspectives. We examine the (Formula presented.) index's ability to predict various physicochemical properties through quantitative structure-property relationship (QSPR) analysis. Additionally, we describe the extremal trees with respect to (Formula presented.). Furthermore, the maximal tree for (Formula presented.) is characterized in relation to a given maximum degree.