k-Power Graphs of Finite Groups

For a finite group $G$ and for a fixed positive integer $k$, $k\ge 2$, the $k$-power graph of $G$ is an undirected simple graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $x^k=y$ or $y^k=x$. In this paper, we investigate some graph parameters such as number of edges, clique number, connectedness, etc. of $k$-power graphs of finite groups. Also find some properties of $k$-power graphs of finite cyclic groups, and finally we present an application