Algebraic connectivity of the second power of a graph

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 $G$.