有限群的增强幂图的λ数
Lambda Number of the enhanced power graph of a finite group
摘要：
有限群$G$的增强幂图是顶点集为$G$且两个不同顶点$x, y$相邻的简单无向图，如果对某个$z \in G$有$x, y \in \langle z \rangle$.图$Γ$的$L( 2,1)$ -标号是$V(Γ)$的整数标号，使得相邻顶点具有至少相差$2$的标号，并且距离$2$的顶点具有至少相差$1$的标号。$λ$ -$Γ$的编号，用$λ(Γ)$表示，是所有$L( 2,1)$ -标记的最小范围。本文研究了群$G$的增强幂图$\mathcal{P}_E(G)$的λ数。本文将文献[22]中关于幂图的λ数的相应结果推广到增强幂图。此外，对于阶为$n$的非平凡单群$G$，我们证明了$λ(\mathcal{P}_E(G)) = n$当且仅当$G$不是阶为$n\geq 3$的循环群.最后，当$G$是有限幂零群时，我们计算了$λ(\mathcal{P}_E(G))$的精确值.
英文摘要：
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.
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