Lambda Numbers of Finite $p$-Groups

An $L(2,1)$-labelling of a finite graph $\Gamma$ is a function that assigns integer values to the vertices $V(\Gamma)$ of $\Gamma$ (colouring of $V(\Gamma)$ by ${\mathbb{Z}}$) so that the absolute difference of two such values is at least $2$ for adjacent vertices and is at least $1$ for vertices which are precisely distance $2$ apart. The lambda number $\lambda(\Gamma)$ of $\Gamma$ measures the least number of integers needed for such a labelling (colouring). A power graph $\Gamma_G$ of a finite group $G$ is a graph with vertex set as the elements of $G$ and two vertices are joined by an edge if and only if one of them is a positive integer power of the other. It is known that $\lambda(\Gamma_G) \geq |G|$ for any finite group. In this paper we show that if $G$ is a finite group of a prime power order, then $\lambda(\Gamma_G) = |G|$ if and only if $G$ is neither cyclic nor a generalized quaternion $2$-group. This settles a partial classification of finite groups achieving the lower bound of lambda number.