The complement of proper power graphs of finite groups

Abstract

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 $u\ne v$ 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.

29 pages, 14 figures, Lemma 4.1 has been added and consequent changes have been made

References in corpus (4)

• Toroidal and projective commuting and non-commuting graphs
• Embeddings of (proper) power graphs of finite groups
• Toroidality and projective-planarity of intersection graphs of subgroups of finite groups
• Quotient graphs for power graphs