Topological Analysis of Zero-Divisor Graphs in Cartesian Products of Commutative Rings Zη,η ∈N: Insight into First Zagreb Indices

In the realm of algorithmic graph theory and network analysis, this study explores the intricate connections between algebraic structures and graph theory by examining zero-divisor graphs derived from Cartesian products of commutative rings, [Formula: see text]. We analyze the zero-divisor graphs [Formula: see text] and [Formula: see text] to obtain explicit formulas for the first Zagreb index, a crucial metric in topological graph theory. These indices provide valuable insights into the structural properties of networks formed by commutative rings, which have applications in various fields, such as robotics, information theory, elliptic curve cryptography, and physics. Our findings not only deepen the understanding of network complexity within algebraic frameworks but also offer new avenues for computational and theoretical research in pure and applied mathematics.