Lambda Number of the enhanced power graph of a finite group

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 $\Gamma$ is an integer labeling of $V(\Gamma)$ 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 $\lambda$-number of $\Gamma$, denoted by $\lambda(\Gamma)$, 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 $\lambda(\mathcal{P}_E(G)) = n$ if and only if $G$ is not a cyclic group of order $n\geq 3$. Finally, we compute the exact value of $\lambda(\mathcal{P}_E(G))$ if $G$ is a finite nilpotent group.