THE METRIC DIMENSION OF ZERO-DIVISOR GRAPHS OF BOOLEAN RINGS

Abstract This paper resolves the previously open problem of determining the metric dimension of the zero-divisor graph $\Gamma (R)$ for the Boolean ring $R = (\mathbb {Z}_2)^n$ . The unique structure of this graph, characterised by its diameter of $3$ and lack of common neighbours, has hindered all standard approaches. We introduce a novel combinatorial method that constructs an explicit resolving set. Consequently, we provide a precise formula for $\dim _m(\Gamma ((\mathbb {Z}_2)^n))$ , closing a notable gap in the literature on metric dimensions of zero-divisor graphs. As an application, we compute the metric dimension of a zero-divisor graph of a ring with a Boolean factor.