Distinguishing number and distinguishing index of natural and fractional powers of graphs

‎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.