M-Polynomial of Product Graphs
This work provides a unified structural description for degree-based topological indices in graph theory, but it is incremental as it extends existing results at the polynomial level.
The paper tackled the problem of computing the M-polynomial for various graph products, developing explicit formulas for seven types of products to describe vertex-degree interactions under graph constructions.
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 Sierpiński 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.