Computing Metric Dimension of Certain Families of Toeplitz Graphs

The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q1, q2, ... , qk} be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q1), r(a, q2), ... , r(a, qk)), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by Tn〈1, t〉 and Tn〈1, 2, t〉, respectively is discussed and proved that it is constant.