Exploring the Exponential Harmonic Index in QSPR Modelling and Extremal Graph Theory

Abstract

Abstract

Exploring the Exponential Harmonic Index in QSPR Modelling and Extremal Graph Theory

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 is significant in both chemistry and mathematics. It is a variant of the Randi´c index, which is widely recognized as one of the most effective molecular descriptors in investigations of structure-property and structure-activity relationships. In comparison to the Randi´c index, the harmonic index exhibits slightly stronger correlations with the physicochemical properties of molecules.

The harmonic index of a graph G, denoted by H(G), is formulated as:

H = H(G) = ∑ vi vj ∈ E(G) 2 / (di + dj)

, where di and dj represent the degrees of the vertices vi and vj, respectively. 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(G) = ∑ vi vj ∈ E(G) e^(2 / (di + dj)).

The exponential harmonic index (EH) is investigated here from both chemical and mathematical perspectives. We examine the EH index’s capability to predict various physicochemical properties through quantitative structure-property relationship (QSPR) analysis. Furthermore, we describe the maximal and minimal trees with respect to the EH index. Further- more, the maximal tree for EH is characterized in relation to a given maximum degree. Finally, we conclude by summarizing our key insights and outlining potential directions for future research.