On the metric dimension of a zero-divisor graph

摘要

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 is an element of (0,1), there exists a finite graph G such that the ratio dim(M)(G)/vertical bar V(H)vertical bar = q, where H is any induced connected subgraph of G. We provide a metric dimension formula for a zero-divisor graph Gamma(R x F-q) and give metric dimension of the zero-divisor graph Gamma(R-1 x R-2 x ... x R-n), where R-1, R-2, ...,R-n are rz finite commutative rings with each having unity 1 and none of R-i, 1 <= i <= n, being isomorphic to the Boolean ring Pi(n)(i=1) Z(2). We discuss the metric dimension of Cartesian product of zero-divisor graphs and show that there exists a zero-divisor graph Gamma(R-1) x Gamma(R-2) such that dim(M)(Gamma(R-1) x Gamma(R-2)) lies between the numbers dim(M)(Gamma(R-1)) and dim(M)(Gamma(R-1)) + 1, where R-1 and R-2 are any two finite commutative rings (not domains) with each having unity 1.