The Algebraic Connectivity of a Graph and its Complement

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$\overline G$isthecomplementof$G$. In this paper, it is shown that $\max \{{\lambda }_2(G),{\lambda }_2(\bar{G})\}\ge 2\mathrm{/}5$.