Lambda number of the power graph of a finite group

Abstract

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.

13 pages, 1 figure