M-Polynomial of Product Graphs

TL;DR

This paper develops explicit formulas for the M-polynomial of various graph products, providing a unified framework to understand degree-based topological indices in complex graph constructions.

Contribution

It introduces new formulas for the M-polynomial of multiple graph products, extending the understanding of degree interactions in graph theory.

Findings

Explicit formulas for M-polynomial of Cartesian, direct, strong, lexicographic, symmetric-difference, disjunction, and Sierpiński products.

Unified structural description of degree interactions in graph products.

Extension of existing degree-based index results at the polynomial level.

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.