Metric dimension of complement of annihilator graphs associated with commutative rings (Q6056619)

For a connected graph $G(V,E)$ a set of vertices $S\subseteq V(G)$ 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 $R$ is the graph whose vertex set is the set of all non-zero zero-divisors of $R$. In this paper the authors study the metric dimension of the complement of the annihilator graph of $R$ and they give some metric dimension formulae for this graph.