The complement of proper power graphs of finite groups

2016/01/14 by Anitha, T., Rajkumar, R., Gagarin, Andrei
#05C10 #05C25 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1601.03683

Abstract

For a finite group G, the proper power graph \mathscrP^*(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 ≠ v and u^m=v or v^m=u for some positive integer m. In this paper, we consider the complement of \mathscrP^*(G), denoted by \mathscrP^*(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.

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