M-polynomial Based Mathematical Formulation of the Hyperbolic Sombor Index

2026-02-16

AI Summary

This study addresses the challenge of efficiently computing the hyperbolic Sombor index (HSO) for chemical structure analysis by introducing, for the first time, the M-polynomial approach to derive a closed-form analytical expression for the HSO index. The proposed method is systematically applicable to standard graphs and representative alkane isomers—including octane, nonane, and decane—significantly enhancing both computational efficiency and the scope of applicability of this topological index. By integrating graph theory, M-polynomial formalism, and visualization techniques, the work not only enables precise calculation of HSO values but also provides intuitive numerical and graphical representations, thereby expanding the potential of the HSO index in characterizing molecular structures in chemical applications.

Technology Category

Search and Optimization: Combinatorial OptimizationKnowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsSearch and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The numerical values extracted from a graph that indicates its topology are called topological indices. A contemporary and efficient method is to compute a graph's topological indices using the graph polynomial that corresponds to it. This method of identifying degree-based topological indices involves the use of the M-polynomial. Very recently, in 2025, the hyperbolic Sombor index (HSO) was proposed and shows its chemical applicability for octane isomers and the structure sensitivity and abruptness for octane, nonane, and decane isomers, respectively. In this work, we establish the closed derivation formula for the above-mentioned index of a graph based on its M-polynomial. Additionally, we use our proposed derivation formula to calculate the hyperbolic Sombor index of a few standard graphs and chemical families. Moreover, we provide the numerical and graphical representations for the M-polynomial and the computed HSO index of the chemical families.

Problem

• M-polynomial
• Hyperbolic Sombor index
• topological indices
• graph polynomial
• chemical graphs

Innovation

• M-polynomial
• Hyperbolic Sombor index
• topological indices
• graph polynomial
• chemical graph theory