The connected-complement normalized Laplacian sum conjecture

Let $G$ be a graph on $n$ vertices, and let $\lambda_2(G)$ denote the second-smallest eigenvalue of its normalized Laplacian. Connected-complement normalized Laplacian conjecture. If $G$ and $G^c$ are both connected, then

$$\lambda_2(G)+\lambda_2(G^c)\geq\frac{2}{\sqrt{n}}.$$

This conjecture proposes a stronger sum bound under the assumption that both a graph and its complement are connected. It is motivated by computational scatterplots and is explicitly identified as an open problem in the paper.