On topological indices and entropy dynamics over zero divisors graphs under cartesian product of commutative rings

Algebraic graph theory is an important area of mathematics that looks into the complex relationships between different algebraic structures and the many features that graphs have. This interdisciplinary field integrates principles from abstract algebra, exploring structures such as rings, fields, and groups, with concepts from graph theory, committed to revealing the properties and topology of graphs. The study of graph theory focuses on elucidating graphs’ features and topological aspects. In the context of this exploration, a graph denoted as G is categorized as a zero-divisor graph solely if the zero-divisors of the modular ring (Formula presented.) form its vertex set. In the absence of this criterion, the graph does not attain the status of a zero-divisor graph. It is noteworthy that the modulo n operation plays a pivotal role in determining the adjacency of two vertices in this network, contingent on whether the product of those vertices yields zero. This study’s scope includes a close examination of specific topological indices designed for various families of zero-divisor graphs. The focus is predominantly on indices such as the first, second, and second modified Zagrebs; the general and inverse general Randics; the third and fifth symmetric divisions; the harmonic and inverse sum indices; and other often overlooked topological indices. Furthermore, we broaden the analysis to encompass various entropies, including the first, second, and third redefined Zagrebs, across various families of zero-divisor graphs. The incorporation of numerical and graphical comparisons in this work aims to provide a more holistic understanding. These comparisons rely on topological indices computed across the previously expounded families of zero-divisor graphs.