论文标题
有限组的增强功率图的lambda数
Lambda Number of the enhanced power graph of a finite group
论文作者
论文摘要
有限组$ g $的增强功率图是一个简单的无向图，其顶点集为$ g $，如果$ x，y \ in \ langle z \ rangle $ in G $中的某些$ z \，则两个不同的顶点$ x，y $相邻。 $ l（2,1）$ - 图$γ$的标签是$ v（γ）$的整数标签，因此相邻的顶点具有至少$ 2 $的标签，而Vertices距离$ 2 $相隔的标签具有至少$ 1 $的标签。 $λ$ number $γ$，用$λ（γ）$表示，是所有$ L（2,1）$标签的最小范围。在本文中，我们研究了组$ g $的增强功率图$ \ Mathcal {p} _e（g）$的lambda编号。本文将lambda数量的功率图数量在[22]中获得的相应结果扩展到增强的功率图。此外，对于非平凡的简单组$ g $的订单$ n $，我们证明$λ（\ Mathcal {p} _e（g））= n $，并且仅当$ g $不是循环的订单$ n \ geq 3 $。最后，如果$ g $是有限的nilpotent组，我们计算$λ（\ mathcal {p} _e（g））$的确切值。
The enhanced power graph of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. An $L( 2,1)$-labeling of graph $Γ$ is an integer labeling of $V(Γ)$ such that adjacent vertices have labels that differ by at least $2$ and vertices distance $2$ apart have labels that differ by at least $1$. The $λ$-number of $Γ$, denoted by $λ(Γ)$, is the minimum range over all $L( 2,1)$-labelings. In this article, we study the lambda number of the enhanced power graph $\mathcal{P}_E(G)$ of the group $G$. This paper extends the corresponding results, obtained in [22], of the lambda number of power graphs to enhanced power graphs. Moreover, for a non-trivial simple group $G$ of order $n$, we prove that $λ(\mathcal{P}_E(G)) = n$ if and only if $G$ is not a cyclic group of order $n\geq 3$. Finally, we compute the exact value of $λ(\mathcal{P}_E(G))$ if $G$ is a finite nilpotent group.
