11 research outputs found
The complement of proper power graphs of finite groups
For a finite group , the proper power graph of is
the graph whose vertices are non-trivial elements of and two vertices
and are adjacent if and only if and or for some
positive integer . In this paper, we consider the complement of
, denoted by . 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.Comment: 29 pages, 14 figures, Lemma 4.1 has been added and consequent changes
have been mad