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 $n in mathbb{N}$‎, ‎the $n$-subdivision of $G$ is a simple graph $G^{frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$‎. ‎The $m^{th}$ 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 $m^{th}$ power of the $n$-subdivision of $G$‎, ‎i.e.‎, ‎$(G^{frac{1}{n}})^m$ or $n$-subdivision of $m$-th power of $G$‎, ‎i.e.‎, ‎$(G^m)^{frac{1}{n}}$‎. ‎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 $n geqslant 3$ with maximum degree $Delta (G)$‎, ‎and for $kgeqslant 2$‎, ‎$D(G^{frac{1}{k}})leqslant lceil sqrt[k]{Delta (G)} rceil$‎. ‎Finally we prove that for $mgeqslant 2$‎, ‎the fractional power of $G$‎, ‎i.e.‎, ‎$(G^{frac{1}{k}})^m$ and $(G^m)^{frac{1}{k}}$ are distinguished‎ ‎ by at most three edge labels‎.