  For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ 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 $u \neq v$ and $u^m=v$ or $v^m=u$ for some
positive integer $m$. In this paper, we consider the complement of
$\mathscr{P}^*(G)$, denoted by ${\overline{\mathscr{P}^*(G)}}$. 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.

Group Theory