  For a graph $G$, let $\lambda_2(G)$ denote its second smallest Laplacian
eigenvalue. It was conjectured that $\lambda_2(G) + \lambda_2(\overline G) \ge
1$, where $\overline G$ is the complement of $G$. In this paper, it is shown
that $\max\{\lambda_2(G), \lambda_2(\overline G)\} \ge 2/5$.
