The Algebraic Connectivity of a Graph and its Complement

B. Afshari, S. Akbari, M. J. Moghaddamzadeh, B. Mohar

Published 2018-06-18Version 1

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 $\max\{\lambda_2(G), \lambda_2(\overline G)\} \ge 2/5$.