论文标题
有关电源图和组的增强功率图的一些新结果
Some New Results Concerning Power Graphs and Enhanced Power Graphs of Groups
论文作者
论文摘要
组$ \ mathbf g $的定向电源图$ \ vec {\ mathcal p}（\ mathbf g）$是带有顶点set $ g $的简单挖掘物，以便$ x \ rightarrow y $如果$ y $是$ x $的电源。 $ \ Mathbf G $的电源图，由$ \ Mathcal P（\ Mathbf G）$表示，是基础简单图。增强功率图$ \ MATHCAL P_E（\ MATHBF G）$ \ MATHBF G $是带有顶点套装$ G $的简单图，其中两个元素如果生成环状子组，则两个元素相邻。 在本文中，事实证明，如果两组具有同形功率图，则它们也具有同构增强功率图。众所周知，任何有限的nilpotent订单组最多可除以两个素数具有完美的增强功率图。我们调查了所有有限群体是否相同的人，并且我们对该问题获得了负面答案。此外，我们证明，对于任何$ n \ geq 0 $和Prime数字$ P $和$ Q $，每组订单$ p^nq $和$ p^2Q^2 $都具有完美的增强功率图。我们还提供了具有完美增强图的对称和替代组的完整表征。
The directed power graph $\vec{\mathcal P}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph of $\mathbf G$, denoted by $\mathcal P(\mathbf G)$, is the underlying simple graph. The enhanced power graph $\mathcal P_e(\mathbf G)$ of $\mathbf G$ is the simple graph with vertex set $G$ in which two elements are adjacent if they generate a cyclic subgroup. In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that any finite nilpotent group of order divisible by at most two primes has perfect enhanced power graph. We investigated whether the same holds for all finite groups, and we have obtained a negative answer to that question. Further, we proved that, for any $n\geq 0$ and prime numbers $p$ and $q$, every group of order $p^nq$ and $p^2q^2$ has perfect enhanced power graph. We also give a complete characterization of symmetric and alternative groups with perfect enhanced graphs.
