Harmonic Mean Cordial Labeling and Topological Descriptors for Graph Characterization and Machine Learning-Based Molecular Property Prediction

Abstract

Graph labeling provides a rigorous mathematical framework for representing structural properties of graphs, while topological descriptors offer numerical measures of graph structure with important applications in chemical graph theory. Existing cordial and harmonic-mean-based labeling approaches primarily focus on establishing labeling properties for specific graph families, with limited integration of harmonic-mean cordial labeling and descriptor development for systematic molecular characterization and quantitative structure–property relationship (QSPR) analysis. This research develops a mathematically rigorous labeling framework and derives topology-sensitive descriptors for characterizing general graphs and molecular structures. First, fundamental graph-theoretic properties and cordial labeling conditions are formulated. A Harmonic Mean Cordial (HMC) labeling is then defined by incorporating harmonic-mean relationships between vertex-associated labels to construct induced edge labels. The existence conditions of HMC labeling are established for selected graph families through lemmas, propositions, and theorem-based proofs. Subsequently, HMC-based topological descriptors are formulated using vertex degrees and harmonic-mean label contributions. Their mathematical properties, bounds, and relationships with conventional degree-based descriptors are analytically investigated. Molecular structures are represented as molecular graphs. Finally, correlation analysis and multiple linear regression-based QSPR modeling are employed to examine the relationship between HMC descriptors and selected physicochemical properties. The analysis establishes HMC labeling conditions and descriptor formulations, while regression analyses evaluate their ability to characterize molecular structures and predict selected properties. Experimental results achieve 0.991 for molar volume, 0.988 for polarizability, 0.742 for PSA, and 0.993 for boiling point. The proposed approach connects graph structure with molecular properties and extends the application of graph-labeling theory to chemical graph theory.