AI 中文总结
本文刻画了具有指数$q$且幂图为友谊图、萤火虫型图或火炬图的有限群，证明幂图为友谊图当且仅当$q=3$，仅$S_3$和$A_4$具萤火虫型幂图，无有限群幂图为火炬图，并确定了相关图类的广义距离谱$D_{\alpha}$谱。
AI 中文摘要
有限群$G$的幂图$P(G)$是顶点集为$G$、边集为$E(P(G))=\{uv: u,v \in G, u \neq v, u \in \langle v \rangle \text{或} v \in \langle u \rangle\}$的图，其中$\langle x\rangle$表示由$x$生成的循环子群。本文刻画了所有具有指数$q$且幂图为友谊图、萤火虫型图或火炬图的有限群，证明了具有指数$q$的有限群$G$的幂图为友谊图当且仅当$q=3$，在交换群情形下等价于$G \cong \mathbb{Z}_3^{n}$；进一步表明，在所有对称群和交错群中，仅$S_3$和$A_4$具有萤火虫型幂图，且不存在有限群的幂图同构于火炬图；最后确定了这些图类的广义距离谱$D_{\alpha}$谱。
英文摘要
The power graph $P(G)$ of a finite group $G$ is the graph with vertex set $G$ and edge set $E(P(G))=\{uv:\ u,v \in G,\ u \neq v,\ u \in \langle v \rangle \ \text{or}\ v \in \langle u \rangle\},$ where $\langle x\rangle$ denotes the cyclic subgroup generated by $x$. In this paper, we characterise all the finite groups with exponent $q$ whose power graphs are friendship graphs, firefly-type graphs, or torch graphs. We prove that the power graph of a finite group $G$ with exponent $q$ is a friendship graph if and only if $q=3$. In particular, in the abelian case, this is equivalent to $G\cong\mathbb{Z}_3^{n}$. We further show that, among all the symmetric and alternating groups, only $S_3$ and $A_4$ have firefly-type power graphs, whereas no finite group has a power graph isomorphic to a torch graph. Finally, we determine the generalised distance spectra $D_α$-spectra of these graph classes.
