Metric dimension of the complement of the zero-divisor graph

Let S be the smallest subset of vertices in a graph G such that every vertex outside of S has a unique distance vector with respect to S. Then |S| is defined as the metric dimension of G and it is denoted by \(dim_M(G)\) $di{m}_M(G)$ . In this paper, the metric dimension of the complement of the zero-divisor graph associated with a commutative ring is discussed. Several formulae for different classes of rings are given.