The complement of proper power graphs of finite groups

For a finite group $G$, the proper power graph ${\mathcal{P}}^{\ast }(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u\mathrm{\ne }v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of ${\mathcal{P}}^{\ast }(G)$, denoted by $\overline{{\mathcal{P}}^{\ast }(G)}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we also determine the diameter and girth of the complement of proper power graphs of finite groups.