交换图是线图的有限群
Finite groups whose commuting graphs are line graphs
AI总结：
本文分类了交换图及其两种变体为线图或线图补图的有限群，给出了完整刻画。
摘要：
群$G$的交换图$\Gamma(G)$是一个简单无向图，其顶点集为群元素，两个元素$x$和$y$相邻当且仅当在$G$中$xy=yx$。通过移除$G$的单位元以及$\Gamma(G)$的所有支配顶点，得到的$\Gamma(G)$的子图分别为$\Gamma^*(G)$和$\Gamma^{**}(G)$。本文分类了所有使得图$\Delta(G) \in \{\Gamma(G), \Gamma^*(G), \Gamma^{**}(G)\}$是某个图的线图的有限群$G$。我们还分类了所有使得图$\Delta(G) \in \{\Gamma(G), \Gamma^*(G), \Gamma^{**}(G)\}$是线图的补图的有限群$G$。
英文摘要：
The commuting graph ${Γ(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $Γ(G)$, the resulting subgraphs of $Γ(G)$ are $Γ^*(G)$ and $Γ^{**}(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the complement of line graph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
