The complement of proper power graphs of finite groups

For a finite group G, the proper power graph 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 6= v and u m = v or v m = u for some positive integer m. In this paper, we consider the complement of P ∗ (G), denoted by 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, outerplanar, toroidal, or projective. Among the other results, we also determine the diameter and girth of the complement of proper power graphs of finite groups