On connectedness of power graphs of finite groups

Abstract

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.