论文标题
在有限组的功率图的差图上
On the Difference Graph of power graphs of finite groups
论文作者
论文摘要
有限组$ g $的功率图是一个简单的无向图，带顶点套装$ g $，如果一个是另一个力量，则两个顶点相邻。有限组$ g $的增强功率图是一个简单的无向图，其顶点集是$ g $，如果存在$ c \ c \ c $ c $，则$ a $ a $和$ b $都是$ c $的powers，则两个vertices $ a $ a $和$ b $相邻。在本文中，我们研究了有限组$ g $的差异图$ \ MATHCAL {d}（g）$，这是增强功率图和$ g $的功率图的差异，所有隔离的顶点已删除。我们研究了有限基团的差异图以及其他结果。我们首先表征一个任意的有限组$ g $，以便$ \ mathcal {d}（g）$是弦图，星形图，可占主导地位，阈值图和拆分图。由此，我们得出的结论是，后四个图表与$ \ Mathcal {d}（g）$相当。通过应用这些结果，我们将nilpotent group $ g $分类为$ \ mathcal {d}（g）$属于上述五个图形类。这表明，当$ g $是nilpotent时，所有这些图形类都是$ \ mathcal {d}（g）$等效的。然后，我们表征其差异图的nilpotent群体是Cophaph，二分，Eulerian，Planar和Outerplanar。最后，我们考虑了非努力组的差异图，并确定$ n $的值，以便对称组$ s_n $和交替的组$ a_n $的差异图是Cophaph，Cophart，Chordal，Split和Threshold。
The power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group $G$ is a simple undirected graph whose vertex set is the group $G$ and two vertices $a$ and $b$ are adjacent if there exists $c \in G$ such that both $a$ and $b$ are powers of $c$. In this paper, we investigate the difference graph $\mathcal{D}(G)$ of a finite group $G$, which is the difference of the enhanced power graph and the power graph of $G$ with all isolated vertices removed. We study the difference graphs of finite groups with forbidden subgraphs among other results. We first characterize an arbitrary finite group $G$ such that $\mathcal{D}(G)$ is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equivalent for $\mathcal{D}(G)$. By applying these results, we classify the nilpotent groups $G$ such that $\mathcal{D}(G)$ belong to the aforementioned five graph classes. This shows that all these graph classes are equivalent for $\mathcal{D}(G)$ when $G$ is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of $n$ such that the difference graphs of the symmetric group $S_n$ and alternating group $A_n$ are cograph, chordal, split, and threshold.
