Metric dimension and strong metric dimension in annihilator-ideal graphs

Let R be a commutative ring with identity and A(R) be the set of ideals with non-zero annihilator. The annihilator-ideal graph of R is defined as the graph \(\mathrm{A_I}(R)\) ${\mathrm{A}}_{\mathrm{I}}(R)$ with the vertex set \(A(R)^*=A(R)\setminus \{0\}\) $A{(R)}^{\ast }=A(R)\backslash \{0\}$ and two distinct vertices L, K are adjacent if and only if \(\textrm{Ann}_R(K) \cup \textrm{Ann}_R(L)\) ${\text{Ann}}_R(K)\cup {\text{Ann}}_R(L)$ is a proper subset of \(\textrm{Ann}_R(KL)\) ${\text{Ann}}_R(KL)$ . In this paper, we determine the metric dimension of \(\mathrm{A_I}(R)\) ${\mathrm{A}}_{\mathrm{I}}(R)$ . Also, the twin-free clique number for \(\mathrm{A_I}(R)\) ${\mathrm{A}}_{\mathrm{I}}(R)$ is computed and as an application the strong metric dimension in annihilator-ideal graphs is given.