Faught–Nordhaus–Gaddum spectral-gap sum conjecture

Let $G$ be a graph on $n$ vertices, and let $\overline{G}$ denote its complement. Assume that both $G$ and $\overline{G}$ are connected; $\operatorname{Gap}(G)$ denotes the difference between the two largest eigenvalues of the transition probability matrix of the random walk on $G$. Faught–Nordhaus–Gaddum sum conjecture.

$$\operatorname{Gap}(G)+\operatorname{Gap}(\overline{G}) \ge \frac{2}{\sqrt{n}}.$$

This is a sum version of the Nordhaus–Gaddum problem for spectral gaps, under the connectivity hypotheses that make both gaps applicable. Its resolution status is not specified in the supplied text.