有限群的幂图的λ数
Lambda number of the power graph of a finite group
摘要：
有限群$G$的幂图$Γ_G$是顶点集为$G$的图，其中两个不同的元素相邻，如果一个是另一个的幂。图$Γ$的$L(2, 1)$ -标号是将标号从非负整数分配给$Γ$的所有顶点，使得距离为2的顶点得到不同的标号，并且相邻顶点得到至少相隔$2$的标号。$Γ$的λ数，用$λ(Γ)$表示，是在$Γ$的所有$L(2, 1)$ -标记上的最小跨度。本文给出了$λ(Γ_G)$的上界，并给出了达到上界的充要条件.作为应用，我们计算了$G$是二面体群、广义四元数群、$\mathcal{P}$ -群或$pq^n$阶循环群时$λ(Γ_G)$的精确值，其中$p$和$q$是不同的素数，$n$是正整数.
英文摘要：
The power graph $Γ_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $Γ$ is an assignment of labels from nonnegative integers to all vertices of $Γ$ such that vertices at distance two get different labels and adjacent vertices get labels that are at least $2$ apart. The lambda number of $Γ$, denoted by $λ(Γ)$, is the minimum span over all $L(2, 1)$-labelings of $Γ$. In this paper, we obtain bounds for $λ(Γ_G)$, and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of $λ(Γ_G)$ if $G$ is a dihedral group, a generalized quaternion group, a $\mathcal{P}$-group or a cyclic group of order $pq^n$, where $p$ and $q$ are distinct primes and $n$ is a positive integer.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
