the co-prime power order graph of a finite group

In this study‎, ‎we generalized the co-prime graph of a finite group called the co-prime power order graph of a finite group‎. ‎it is denoted by $beta_g$‎, ‎and its vertex set is $g$‎, ‎such that two distinct vertices $x$ and $y$ are adjacent if and only if $operatorname{gcd}(|x|,|y|)=p^n$‎, ‎where $p$ is a prime number‎, ‎and $n in mathbb{z}^{+} cup{0}$‎. ‎we characterized complete graphs and planar graphs on the co-prime power order graphs‎, ‎and investigated some properties of graph $beta_g$ for some groups such as cyclic groups‎, ‎dihedral groups‎, ‎and the generalized quaternion groups‎, ‎and obtained the vertex-connectivity among them‎. ‎finally‎, ‎we characterized some induced subgraphs of co-prime power order graph for some finite groups.