© 2019, IJSRMSS All Rights Reserved 74 International Journal of Scientific Research in ______________________________ Research Paper . Mathematical and Statistical Sciences Vol.6, Issue.6, pp.74-78, December (2019) E-ISSN: 2348-4519 Metric Dimension of Zero-Divisor Graph for the Ring Z n Mithun Basak 1* , Laxman Saha 2 , Kalishankar Tiwary 3 1,2 Dept. of Mathematics, Balurghat College, Balurghat 733101, India 3 Dept. of Mathematics, Raiganj University, Raiganj-733134, India *Corresponding Author: basak.mithun5007@gmail.com, Available online at: www.isroset.org Received: 25/Nov/2019, Accepted: 05/Dec/2019, Online: 31/Dec/2019 Abstract- Metric Dimension of a simple connected graph is the minimum number of vertices those are used to identify each vertex of the graph uniquely using distance code. In this paper, we determine metric dimension of zero-divisor graph associated with the ring n Z . Keywords: Ring, Metric Dimension, Zero divisors, Zero-divisor Graph. I. INTRODUCTION Zero-Divisor graphs were first introduced by Istvan Beck in [7] to give colors in a commutative ring. However, he took all elements of a ring R be vertices of the graph. In a paper [4] in 1999, Anderson and Livingston modified it by taking only nonzero zero divisors of a ring as vertices. They studied the interplay between the ring-theoretic properties of a commutative ring and the graph theoretic properties of its zero-divisor graph. Many properties of zero-divisor graph of a commutative ring has been studied in [1, 2, 3, 4, 5, 14, 15, 16]. The study of zero divisor graph was extended to non commutative ring in [18] by Redmond and an ideal- based zero divisor graph zero divisor graph of a commutative ring by Redmond. Navigation problem can be studied in terms of graph theoretic model. Suppose, a Robot, navigating in a space want to determine its location uniquely. Now the question is how can a robot determine its positions uniquely ? One way to determine the position is the way of calculating its distances from stations. Again still a problems that how many stations should be considered to determine the positions of a robot uniquely. Thus it important to determine the minimum number of stations and their positions such that the problem of locating each robot exactly can be easily solved out. This problem can be transformed into graph theory if we treat each station as a vertex and between two station there is a edge if a robot moves between these two stations. Then the main problem is transformed into finding the minimum number of vertices to determine each vertex uniquely using graph theoretic distances. In literature, these types of problems are known as metric dimension of graphs. The concept of the metric dimension of a graph was first introduced by Slater [20] and independently by Harary and Melter [12]. Their introduction of this invariant was motivated by its application to the placement of a minimum number of sonar/loran detecting devices in a network so that the position of every vertex in the network can be uniquely described in terms of its distances to the devices in the set. Throughout this paper, (, ) G VE  denotes a simple connected graph and (,) duv represents the distance between two vertices u and v in G. For an ordered subset 1 2 { , ,..., } ( ) k S ww w VG   and a vertex v of G , the code of v with respect to S is a k -vector given by 1 2 () ((, ), (, ), ...., (, )) S k code v dvw dvw dvw  If () () S S code u code v  implies that u v  for all pairs , uv of vertices of G , then S is called a resolving set for the graph G . The metric dimension of a graph G , is the minimum cardinality of a resolving set for G and it is denoted by ( ) G  . The metric dimension has been extensively studied by various authors in last few decades. The metric dimension has been discovered or invented in different forms and appeared in different application including combinatorial optimization [19], strategies for the master mind game [9], network discovery and verification [8], robot navigation [13] and so on. It is stated in [11, 13] that finding metric dimension of a graph is NP- complete problem. In this article, we give a lower bound of metric dimension for