Computing Metric Dimension of Power of Total Graph

For a connected graphG, the distance$d(u, v)$between any two verticesu,v~∈ ~V(G)is defined as;d(u,v)=min _Pu,v of~Pᵤᵥ.e the minimum length of path connecting any two vertices. Two verticesu,vof a graphGare said to be resolved by a verticesWof a graphGifd(u,w) ≠ d(v,w). An ordered setW = ₁,w₂,w₃,…,wₖ\\,\,∈ \,\,V(G)is resolving set for a connected graphG, if for any two verticesu,vthere existswᵢ ∈ Wsuch thatd(u,wᵢ) ≠ d(v, wᵢ). The representation of verticesvw.r.tWis represented byr(v|W)which isk-vector(k-tuple)(d(v,w₁),d(v,w₂),d(v,w₃),…,d(v,wₖ)). If the representation of all the vertices of graphGare different w.r.tWthenWis the resolving set forG[1]. A resolving set that contains minimum numbers of vertices ismetric basisforG. The cardinality of smallest resolving set is themetric dimensionofG, represented bydim(G)[2],[3]. A family of connected graphGhas unbounded metric dimension ifdim(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.