Analytical, structural and extremal properties of degree-based indices on Cartesian graph products

We develop a unified discrete framework for computing and analyzing degree-based topological indices of Cartesian graph products, a class of constructions that plays a central role in applied and structural graph theory. Our approach builds on the M -polynomial and its operator calculus, enabling the derivation of closed-form expressions for an arbitrary product G 1 □ ⋯ □ G ℓ . This formulation yields exact evaluations for a broad family of classical indices, including the First and Second Zagreb indices and the Forgotten index, while avoiding product-wise combinatorial enumeration. A second component of the work establishes sharpened bounds for degree-based indices on trees. These bounds refine earlier estimates and are obtained through a combination of discrete analytic techniques and structural constraints on the degree sequences. We further introduce a general inheritance principle that describes how the extremal behavior of indices is preserved under Cartesian multiplication. Using tools from majorization and discrete convexity, we prove that graphs extremal within their classes remain extremal after an arbitrary number of Cartesian factors, yielding systematic bounds for large composite structures. Finally, we perform a numerical investigation of the indices M 1 and M 2 in random families of Cartesian products. The experiments provide empirical evidence suggesting approximately linear growth of the mean and quadratic growth of the variance, and suggest that a log-normal model provides an accurate approximation for large graph sizes.