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

Abstract: 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 $G$.