Some Properties of Proper Power Graphs in Finite Abelian Groups

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