On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings

Given a finite commutative unital ring S having some non-zero elements x, y such that x.y=0, the elements of S that possess such property are called the zero divisors, denoted by ZS. We can associate a graph to S with the help of zero-divisor set ZS, denoted by ? S (called the zero-divisor graph), to study the algebraic properties of the ring S. In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of S. To do so, we will discuss the zero-divisor graphs for the ring of integers DOUBLE-STRUCK CAPITAL Z(m) modulo m, some quotient polynomial rings, and the ring of Gaussian integers DOUBLE-STRUCK CAPITAL Z(m)i modulo m. Then, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of ? S. In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.