For a graph $G$, let $λ_2(G)$ denote its second smallest Laplacian eigenvalue. It was conjectured that $λ_2(G) + λ_2(\overline{G}) \geq 1$, where $\bar{G}$ is the complement of $G$. Here, we prove this conjecture in the general case. Also, we will show that $\max\{λ_2(G), λ_2(\overline{G})\} \geq 1 - O(n^{-\frac 13})$, where $n$ is the number of vertices of $G$.
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