论文标题
在组的增强功率图上
On the enhanced power graph of a group
论文作者
论文摘要
组$ g $的增强功率图$ \ MATHCAL {p} _e（g）$是带顶点套装$ g $的图形，如果它们属于同一环状亚组，则两个顶点相邻。在本文中，我们考虑了有限组的最低程度，独立性数量和增强功率图的匹配数。我们首先研究这些图形不变的$ \ Mathcal {p} _e（g）$当$ g $是任何有限组时，然后确定$ g $是有限的Abelian $ p $ -p $ -group，$ u_ {6n} = \ langle a，langle a，b：a^= b：a^{2n} = b^3 3 = e，dih = e， $ d_ {2n} $，或半二面组$ sd_ {8n} $。如果$ g $是这些组中的任何一个，我们证明$ \ Mathcal {p} _e（g）$是完美的，然后获得其强度的度量尺寸。此外，我们为任何有限的Abelian Group $ g $的$ \ Mathcal {p} _e（g）$的独立数提供了一个表达式。这些结果以及某些已知的相等性产生边缘连接性，顶点覆盖数和边缘覆盖相应组的增强功率图的数量。
The enhanced power graph $\mathcal{P}_e(G)$ of a group $G$ is a graph with vertex set $G$ and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number and matching number of enhanced power graphs of finite groups. We first study these graph invariants for $\mathcal{P}_e(G)$ when $G$ is any finite group, and then determine them when $G$ is a finite abelian $p$-group, $U_{6n} = \langle a, b : a^{2n} = b^3 = e, ba =ab^{-1} \rangle$, the dihedral group $D_{2n}$, or the semidihedral group $SD_{8n}$. If $G$ is any of these groups, we prove that $\mathcal{P}_e(G)$ is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of $\mathcal{P}_e(G)$ for any finite abelian group $G$. These results along with certain known equalities yield the edge connectivity, vertex covering number and edge covering number of enhanced power graphs of the respective groups as well.
