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 = {q
₁
, q
₂
, ... , q
_k
} 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, q
₁
), r(a, q
₂
), ... , r(a, q
_k
)), 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 T
_n
〈1, t〉 and T
_n
〈1, 2, t〉, respectively is discussed and proved that it is constant.