A Nordhaus--Gaddum problem for the spectral gap of a graph

Let $G$ be a graph on $n$ vertices, with complement $\bar{G}$. 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 $\bar{G}$ is $\mathrm{\Omega }(1\mathrm{/}n)$. Moreover, if all degrees are $\mathrm{\Omega }(n)$ and $n-\mathrm{\Omega }(n)$, then the larger spectral gap of $G$ and $\bar{G}$ is $\mathrm{\Theta }(1)$. We also show that if the maximum degree is $n-O(1)$ or if $G$ is a join of two graphs, then the spectral gap of $G$ is $\mathrm{\Omega }(1\mathrm{/}n)$. Finally, we provide a family of connected graphs with connected complements such that the larger spectral gap of $G$ and $\bar{G}$ is $O(1\mathrm{/}{n}^{3\mathrm{/}4})$.