  For a graph $G$, let $\lambda_2(G)$ denote its second smallest Laplacian
eigenvalue. It was conjectured that $\lambda_2(G) + \lambda_2(\overline{G})
\geq 1$, where $\bar{G}$ is the complement of $G$. Here, we prove this
conjecture in the general case. Also, we will show that $\max\{\lambda_2(G),
\lambda_2(\overline{G})\} \geq 1 - O(n^{-\frac 13})$, where $n$ is the number
of vertices of $G$.
