On the metric dimension of a zero-divisor graph

Let R be a commutative ring with unity 1 and let G(V,E) be a simple graph. In this research article, we study the metric dimension in zero-divisor graphs associated with commutative rings. We show that for a given rational q∈(0,1), there exists a finite graph G such that the ratio dimM(G)|V(H)|=q, where H is any induced connected subgraph of G. We provide a metric dimension formula for a zero-divisor graph Γ(R×𝔽q) and give metric dimension of the zero-divisor graph Γ(R1×R2×⋯×Rn), where R1,R2,…,Rn are n finite commutative rings with each having unity 1 and none of Ri, 1≤i≤n, being isomorphic to the Boolean ring ∏i=1nℤ2. We discuss the metric dimension of Cartesian product of zero-divisor graphs and show that there exists a zero-divisor graph Γ(R1)×Γ(R2) such that dimM(Γ(R1)×Γ(R2)) lies between the numbers dimM(Γ(R1)) and dimM(Γ(R1))+1, where R1 and R2 are any two finite commutative rings (not domains) with each having unity 1.