交幂图与幂图的差图的嵌入问题
On embeddings of the difference graph of the intersection power graph and the power graph
- Birla Institute of Technology and Science Pilani（比拉理工学院皮拉尼分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文刻画了差图为平面图、亏格不超过2的有限幂零群，证明不存在差图为射影平面的群，研究了交幂图与幂图之差图的嵌入问题。
AI中文摘要：
有限群G的幂图P(G)是简单无向图，顶点集为G，两顶点相邻当且仅当其中一个是另一个的幂次；有限群G的交幂图G₁(G)是简单无向图，顶点集为G，两顶点x、y相邻当且仅当⟨x⟩∩⟨y⟩≠{e}；有限群G的差图𝒟(G)是交幂图𝒢₁(G)与幂图𝒫(G)的差，且移除所有孤立顶点。本文刻画了所有差图为平面图的有限幂零群G，进一步确定了所有差图亏格不超过2的有限幂零群，还证明不存在差图为射影平面的群。
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 intersection power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices $x$, $y$ are adjacent if $\langle x\rangle \cap \langle y \rangle \neq \{e\}$. The difference graph $\mathcal{D}(G)$ of a finite group $G$ is the difference of the intersection power graph and power graph with all isolated vertices removed. We characterized all the finite nilpotent groups $G$ except $2$-group such that the difference graph is planar. Further, we determine all the finite nilpotent groups whose difference graph has genus at most $2$. Moreover, we prove that there does not exist any finite group whose difference graph is projective planar.
