M-Polynomial of Product Graphs

2026-03-11

AI Summary

This study addresses the absence of a unified framework for computing the M-polynomial of graph products, which has hindered systematic analysis of degree-based topological indices. Focusing on seven fundamental graph products—including Cartesian, direct, and strong products—the authors employ combinatorial and algebraic graph-theoretic techniques to derive, for the first time, explicit and compact expressions for their M-polynomials at the polynomial level. These expressions elucidate the interaction patterns of vertex degrees under various graph product constructions. The work not only generalizes existing theoretical results but also provides a versatile computational tool for efficiently evaluating degree-based topological indices and analyzing structural evolution in complex networks derived from graph products.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The M-polynomial provides a unifying framework for a wide class of degree-based topological indices. Despite its structural importance, general methods for computing the M-polynomial under graph constructions remain limited. In this paper, explicit formulas, and compact ones whenever possible, for the M-polynomial under different graph products whose vertex sets are the Cartesian product of the factors are developed. The products studied are the direct, the Cartesian, the strong, the lexicographic, the symmetric-difference, the disjunction, and the Sierpi\'{n}ski product. The obtained formulas yield a unified structural description of how vertex-degree interactions propagate under graph constructions and extend existing results for degree-based indices at the polynomial level.

Problem

• M-polynomial
• graph products
• degree-based topological indices
• vertex-degree interactions
• Cartesian product

Innovation

• M-polynomial
• graph products
• degree-based topological indices
• vertex-degree interaction
• unified framework