Superpower graphs of finite groups

For a finite group [Formula: see text] the superpower graph [Formula: see text] of [Formula: see text] is an undirected simple graph with vertex set [Formula: see text] and two vertices are adjacent in [Formula: see text] if and only if the order of one divides the order of the other in [Formula: see text]. The aim of this paper is to provide tight bounds for the vertex connectivity, discuss Hamiltonian-like properties of superpower graph of finite non-Abelian groups having an element of exponent order. We also give some general results about superpower graphs and their relation to other graphs such as the Gruenberg–Kegel graph.