  For a finite group $G$, the power graph $\Gamma_G$ is a graph whose set of
vertices is equal to $G$ and two distinct elements of $G$ are adjacent if and
only if one of them is a positive integer power of the other. An
$L(2,1)$-labelling of $\Gamma_G$ is an integer-valued function defined on $G$
so that the distance between the images of two adjacent vertices (resp.
vertices with distance two) are at least two (resp. at least one). The lambda
number $\lambda(G)$ is defined to be the least difference between the largest
and the smallest integer values assigned to the vertices of $\Gamma_G$ for all
possible $L(2,1)$-labellings of $\Gamma_G$. It is known that $\lambda(G) \geq
|G|$. In this paper, we prove that if $G$ is a finite simple group, then
$\lambda(G) = |G|$ except when $G$ is cyclic of prime order. This settles a
partial classification of finite groups $G$ that achieve the lower bound on of
lambda number.
