  Denote the Laplacian of a graph $G$ by $L(G)$ and its second smallest
Laplacian eigenvalue by $\lambda_2(G)$. If $G$ is a graph on $n\ge 2$ vertices,
then it is shown that the second smallest eigenvalue of $L(G) + \frac{1}{n}
L(\overline{G^2})$ is at least 1, where $\overline{G^2}$ is the complement of
the second power of $ G $. As a corollary of this result, it is shown that
\begin{itemize}
  \item $ n \, \lambda_2(G) \ge \lambda_2(G^2), $
  \item $ \lambda_2(G) \ge 1-\frac{|D_G|}{n}, $
  \item $ \lambda_2(G) + \lambda_2(\Gb) \ge 1, $ \end{itemize} where $|D_G|$ is
the number of vertices of eccentricity at least 3 in $G$.
