论文标题
有限组的增强功率图的补充
The complement of enhanced power graph of a finite group
论文作者
论文摘要
有限的组$ g $的增强功率图$ \ MATHCAL {p} _e（g）$是简单的无向图，其顶点集为$ g $，而两个不同的顶点$ x，y $相邻，如果$ x，y \ in \ langle z \ rangle z \ rangle $ in g $ in g $ in g $。在本文中，我们给出了卡梅伦[6]提出的问题的肯定答案，该问题指出：不可行的功率图$ \ bar $ \ bar {\ mathcal {p} _e（g）} $的补充是非循环$ g $的$ g $与隔离端口相互连接的组件只有一个连接的组件吗？我们对所有有限组$ g $进行分类，以使图$ \ bar {\ Mathcal {p} _e（g）} $是双分部分。我们表明图形$ \ bar {\ mathcal {p} _e（g）} $是弱完美的。此外，我们研究了$ \ bar {\ bar {\ mathcal {p} _e（g）} $的子图$ \ bar {\ Mathcal {p} _e（g^*）} $。我们对所有有限组$ g $进行了分类，以使该图为$ \ bar {\ mathcal {p} _e（g^*）} $是单轮和五囊。我们证明了有限组$ g $的不存在，因此图形$ \ bar {\ mathcal {p} _e（g^*）} $是bicyclic，tricyclic或tetracyclic。最后，我们表征所有有限的$ g $，以便分别是外部平台，平面，投射 - 平面和圆环。
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.
