For a connected graph G(V, E) a set of vertices \(S\subseteq V(G)\) resolves the graph G, and S is a resolving set of G, if every vertex is uniquely determined by its vector of distances to the vertices of S. A resolving set S of minimum cardinality is a metric basis for G, and the number of elements in the resolving set of minimum cardinality is the metric dimension of G. Let R be a commutative ring with non-zero identity. The annihilator graph of R, denoted by AG(R), is the (undirected) graph whose vertex set is the set of all non-zero zero-divisors of R and two distinct vertices x and y are adjacent if and only if \(ann_R(xy)\ne ann_R(x)\cup ann_R(y)\). In this paper, the metric dimension of the complement of AG(R) is studied and some metric dimension formulae for this graph are given.

对于连通图G ( V ,  E )，一组顶点\(S\subseteq V(G)\)解析图G，并且S是G的解析集 ，如果每个顶点都由其距离向量唯一确定到S的顶点。最小基数的解析集合S是G的度量基础，最小基数的解析集合中元素的个数就是G的度量维数。令R是一个具有非零身份的交换环。R的湮灭图，表示为 AG ( R ) 是（无向）图，其顶点集是R的所有非零零因数的集合，并且两个不同的顶点x和y相邻当且仅当\(ann_R(xy)\ne ann_R( x)\cup ann_R(y)\)。本文研究了AG ( R )的补集的度量维数，并给出了该图的一些度量维数公式。