THE METRIC DIMENSION OF ZERO-DIVISOR GRAPHS OF BOOLEAN RINGS

Source abstract

Abstract This paper resolves the previously open problem of determining the metric dimension of the zero-divisor graph Γ ( R ) $\Gamma (R)$ normal upper Gamma left parenthesis upper R right parenthesis for the Boolean ring R = ( Z 2 ) n $R = (\mathbb {Z}_2)^n$ upper R equals left parenthesis double struck upper Z 2 right parenthesis Superscript n . The unique structure of this graph, characterised by its diameter of 3 $3$ 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 ⁡ ( Γ ( ( Z 2 ) n ) ) $\dim _m(\Gamma ((\mathbb {Z}_2)^n))$ dimension Subscript m Baseline left parenthesis normal upper Gamma left parenthesis left parenthesis double struck upper Z 2 right parenthesis Superscript n Baseline right parenthesis right parenthesis , 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.