M-Polynomial of Product Graphs

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.