Abstract
For a connected graph G, the distance d(u, v) between any two vertices u, v is an element of V(G) is defined as; d(u, v) = min(Pu,v){length of P-uv} i.e the minimum length of path connecting any two vertices. Two vertices u, v of a graph G are said to be resolved by a vertices W of a graph G if d(u, w) not equal d(v, w). An ordered set W = {w(1), w(2), W-3..., w(k)} is an element of V(G) is resolving set for a connected graph G, if for any two vertices u, v there exists w(i), is an element of W such that d(u, w(i)) not equal d(v, w(i)). The representation of vertices v w.r.t W is represented by r(v vertical bar W) which is k-vector(k-tuple) (d(v, w(1)), d(v, w(2)), d(v, w(3)), ..., d(v, w(k))). If the representation of all the vertices of graph G are different w.r.t W then W is the resolving set for G [1]. A resolving set that contains minimum numbers of vertices is metric basis for G. The cardinality of smallest resolving set is the metric dimension of G, represented by dim(G) [2], [3]. A family of connected graph G has unbounded metric dimension if dim(G) depends on order of a graph. In this paper, we show that total graph of path power three and four have unbounded metric dimension. We also proved some results on edges of power of path graph.