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

The distinguishing number (index) $D(G)$ ($D^{\mathrm{\prime }}(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. For any $n \in $\mathbb{N}$$,the$n$-subdivisionof$G$isasimplegraph$G^{\frac{1}{n}}$whichisconstructedbyreplacingeachedgeof$G$withapathoflength$n$. 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. The fractional power of $G$, denoted by $G^{\frac{m}{n}}$ is $m^{th}$ power of the $n$-subdivision of $G$ or $n$-subdivision of $m$-th power of $G$. In this paper we study the distinguishing number and distinguishing index of natural and fractional powers of $G$. We show that the natural powers more than two of a graph distinguished by three edge labels. Also we show that for a connected graph $G$ of order $n⩾3$ with maximum degree $\Delta (G)$,$D(G^{\frac{1}{k}})\leqslant min{s: 2^k+\sum^s_{n=3}n^{k-1}\geqslant \Delta (G)}$andfor$m\geqslant 3$,$D’(G^{\frac{m}{k}})\leqslant 3$.