有限群$G$的增强幂图$\mathcal{P}_E(G)$是一个简单无向图，它的顶点集是$G$，如果对某个$z \in G$有$x, y \in \langle z \rangle$，则两个不同的顶点$x, y$是相邻的.本文肯定地回答了Cameron [6]提出的问题：非循环群$G$的增强幂图$\bar{\mathcal{P}_E(G)}$的补图除了孤立点外是否只有一个连通分支？我们对所有有限群$G$进行分类，使得图$\bar{\mathcal{P}_E(G)}$是二部图。证明了图$\bar{\mathcal{P}_E(G)}$是弱完美图.进一步，我们研究了由$\bar{\mathcal{P}_E(G)}$的所有非孤立点诱导的$\bar{\mathcal{P}_E(G)}$的子图$\bar{\mathcal{P}_E(G^*)}$.我们对所有有限群$G$进行分类，使得图是$\bar{\mathcal{P}_E(G^*)}$是单圈和五圈的.证明了不存在有限群$G$使得图$\bar{\mathcal{P}_E(G^*)}$是双环、三环或四环的.最后，我们刻画了所有有限群$G$，使得图$\bar{\mathcal{P}_E(G^*)}$分别是外平面的、平面的、射影平面的和环面的。
英文摘要：
The enhanced power graph $\mathcal{P}_E(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we give an affirmative answer of the question posed by Cameron [6] which states that: Is it true that the complement of the enhanced power graph $\bar{\mathcal{P}_E(G)}$ of a non-cyclic group $G$ has only one connected component apart from isolated vertices? We classify all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G)}$ is bipartite. We show that the graph $\bar{\mathcal{P}_E(G)}$ is weakly perfect. Further, we study the subgraph $\bar{\mathcal{P}_E(G^*)}$ of $\bar{\mathcal{P}_E(G)}$ induced by all the non-isolated vertices of $\bar{\mathcal{P}_E(G)}$. We classify all finite groups $G$ such that the graph is $\bar{\mathcal{P}_E(G^*)}$ is unicyclic and pentacyclic. We prove the non-existence of finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is outerplanar, planar, projective-planar and toroidal, respectively.
Certain properties of the enhanced power graph associated with a finite group
有限群$G$的增强幂图，记作$\mathcal{P}_E(G)$，是顶点集为$G$且两个不同顶点$x, y$相邻的简单无向图，如果对某个$z \in G$有$x, y \in \langle z \rangle$.在这篇文章中，我们确定了所有有限群，使得$\mathcal{P}_E(G)$的最小度和点连通度相等。此外，我们分类的所有群体的（适当的）增强功率图是强正则的。进一步，得到了与某些幂零群相关的增强幂图的点连通性。最后，我们得到了$\mathcal{P}_E(G)$的Wiener指标的一个下界和一个上界，其中$G$是幂零群.达到这些界的有限幂零群幂零群也被刻画了。
On the structure of the power graph and the enhanced power graph of a group
设$G$是一个群。$G$的\emph{power graph}是顶点集为$G$的图，当一个元素是另一个元素的幂时，两个元素之间有一条边。刻画了幂图具有有限独立数的幂零群群。对于有界指数群，证明了它的幂图是完美图，并确定了它的团/色数。进一步证明了对任何群$G$，$G$的幂图的团数至多是可数无限的.我们还通过引入一个位于两者之间的新图来测量功率图与\emph{commuting graph}的接近程度。我们称这个新的图为\emph{enhanced power graph}。对于这三个图的任意一对，我们刻画了使得这对图相等的有限群。
群$G$的增强幂图$\mathcal{P}_e(G)$是顶点集为$G$且两个顶点属于同一循环子群的图。本文研究了有限群的增强幂图的最小度、独立数和匹配数。本文首先研究了当$G$是任意有限群时$\mathcal{P}_e(G)$的这些图不变量，然后确定了当$G$是有限阿贝尔$p$ -群、$U_{6n} = \langle a, b : a^{2n} = b^3 = e, ba =ab^{-1} \rangle$、二面体群$D_{2n}$或半二面体群$SD_{8n}$时的这些图不变量.如果$G$是这些群中的任意一个，我们证明了$\mathcal{P}_e(G)$是完美的，从而得到了它的强度量维数。此外，我们还给出了任意有限交换群$G$的$\mathcal{P}_e(G)$的独立数的表达式.这些结果与某些已知的等式一起给出了相应群的增强幂图的边连通度、顶点覆盖数和边覆盖数。
