The complement of proper power graphs of finite groups
Abstract
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.