主要研究幂零群、内幂零群以及内交换群幂图的相关图论性质. 一般地, 给出有限群 $G$ 的幂图 ${\mathscr{P}}(G)$ 为某图的线图当且仅当 $G$ 为素数幂阶循环群, 得到幂零群与内交换群幂图独立数取临界值时的充要条件, 以及内幂零 群与内交换群幂图可平面化的充要条件. 最后, 分析内幂零群与内交换群真幂图的连通性, 给出了连通情形的直径估计以及非连通情形的连通分支个数.

This paper studies properties of power graphs of nilpotent groups, inner nilpotent groups and inner abelian groups. In general, a power graph ${\mathscr{P}}(G)$ of a finite group $G$ is a line graph if and only if $G$ is a cyclic group of a prime power order. In addition, the necessary and sufficient conditions of power graphs for independent numbers of nilpotent groups and inner abelian groups that take the critical value and the planarization of inner nilpotent groups and inner abelian groups are obtained. Finally, the connectivity of proper power graphs of inner nilpotent groups and inner abelian groups are discussed. This paper gets the diameter estimations and the numbers of connected components under the conditions of connected and disconnected proper power graphs, respectively.