Abstract :
The distinguishing number (resp. index) D(G) (D′(G)) of a graph G is the least integer d such that G has an vertex labeling (resp. edge labeling) with d labels that is preserved only by a trivial automorphism. For any n∈N, the n-subdivision of G is a simple graph G1n which is constructed by replacing each edge of G with a path of length n.The mth power of G, is a graph with same set of vertices of G and an edge between two vertices if and only if there is a path of length at most m between them in G. The fractional power of G, is the mth power of the n-subdivision of G, i.e., (G1/n)m or n-subdivision of m-th power of G, i.e., (Gm)1n. In this paper we study the distinguishing number and the distinguishing index of the natural and the fractional powers of G. We show that the natural powers more than one of a graph are distinguished by at most three edge labels. We also show that for a connected graph G of order n⩾3 with maximum degree Δ(G), and for k⩾2, D(G1k)⩽⌈k√Δ(G)⌉. Finally we prove that for m⩾2, the fractional power of G, i.e., (G1/k)m and (Gm)1/k are distinguished by at most three edge labels.
