A Nordhaus–Gaddum Problem for the Spectral Gap of a Graph

Let $G$ be a graph on $n$ vertices, with complement $\overline{G}$. The spectral gap of the transition probability matrix of a random walk on $G$ is used to estimate how fast the random walk becomes stationary. We prove that the larger spectral gap of $G$ and $\overline{G}$ is $\Omega \left(1∕n\right)$. Moreover, if all degrees are $\Omega \left(n\right)$ and $n-\Omega \left(n\right)$, then the larger spectral gap of $G$ and $\overline{G}$ is $\Theta \left(1\right)$. We also show that if the maximum degree is $n-O\left(1\right)$ or if $G$ is a join of two graphs, then the spectral gap of $G$ is $\Omega \left(1/n\right)$. Finally, we provide a family of connected graphs with connected complements such that the larger spectral gap of $G$ and $\overline{G}$ is $O\left(1/n^{3∕4}\right)$.