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For a graph $G$, let $\lambda_2(G)$ denote its second smallest Laplacianeigenvalue. It was conjectured that $\lambda_2(G) + \lambda_2(\overline{G})\geq 1$, where $\bar{G}$ is the complement of $G$. Here, we prove thisconjecture 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 numberof vertices of $G$.

Proof of a conjecture on the algebraic connectivity of a graph and its complement