Computing dominating number and dominant metric dimension for zero divisor graphs of order at most 10 of small finite commutative rings

In this article we compute the dominating number (DN) and dominant metric dimension (Ddim) of zero divisor graphs of some small finite commutative rings with order not exceeding 14. Consider a commutative ring denoted as and let represent its zero-divisor graph (ZD-graph). The vertices of these graphs correspond to the non-zero divisors (ZD) within the commutative ring (CR), where an edge connects two distinct vertices if their product in the ring results in zero. This paper focuses on studying the domination number and dominant metric dimension for zero divisor graphs of orders 3, 4, 5, 6, 7, 8, 9, and 10 within a small finite commutative ring with a unity. Employing a combination of computational methods and mathematical techniques, our research sheds light on the structural nuances of these small commutative rings, enhancing our comprehension of their algebraic behavior and paving the way for potential applications in algebraic theory and related fields.