关于有限群的幂图的差图
On the Difference Graph of power graphs of finite groups
摘要：
有限群$G$的幂图是一个简单的无向图，其顶点集为$G$，且如果一个顶点是另一个顶点的幂，则两个顶点相邻。有限群$G$的增强幂图是一个简单的无向图，其顶点集是群$G$，并且如果存在$c \in G$使得$a$和$b$都是$c$的幂，则两个顶点$a$和$b$相邻.本文研究了有限群$G$的差图$\mathcal{D}(G)$，它是有限群$G$的增强幂图与去掉孤立点后的幂图的差.研究了具有禁止子图的有限群的差图。我们首先刻画了任意有限群$G$，使得$\mathcal{D}(G)$是弦图、星图、支配图、阈值图和分裂图.由此，我们得出结论，后四个图类是等价的$\mathcal{D}(G)$。应用这些结果，我们对幂零群$G$进行了分类，使得$\mathcal{D}(G)$属于上述五个图类.这表明当$G$是幂零的时，所有这些图类对于$\mathcal{D}(G)$是等价的。然后，我们刻画了差图为上图、偶图、欧拉图、平面图和外平面图的幂零群。最后，我们考虑了非幂零群的差图，并确定了$n$的值，使得对称群$S_n$和交错群$A_n$的差图是余图、弦图、分裂图和阈值图。
英文摘要：
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.
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