On the Strong Metric Dimension of Annihilator Graphs of Commutative Rings

For a connected graph G(V, E), a vertex $$w\in V(G)$$ strongly resolves two vertices $$u, v \in V(G)$$ if there exists a shortest $$u-w$$ path containing v or a shortest $$v-w$$ path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is a simple graph with the vertex set $$Z(R)^*=Z(R){\setminus }\{0\}$$ , 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, we study the strong metric dimension of annihilator graphs associated with commutative rings and some strong metric dimension formulae for annihilator graphs are given.