Some properties of proper power graphs in finite Abelian groups

The power graph of a group [Formula: see text], denoted as [Formula: see text], constitutes a simple undirected graph characterized by its vertex set [Formula: see text]. Specifically, vertices [Formula: see text] exhibit adjacency exclusively if [Formula: see text] belongs to the cyclic subgroup generated by [Formula: see text] or vice versa. The corresponding proper power graph of [Formula: see text] is obtained by taking [Formula: see text] and removing a vertex corresponding to the identity element, which is denoted as [Formula: see text]. In the context of finite abelian groups, this paper establishes the sufficient and necessary conditions for the proper power graph’s connectedness. Moreover, a precise upper bound for the diameter of [Formula: see text] in finite abelian groups is provided with sharpness. This paper also explores the study of vertex connectivity, center, and planarity.