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

Let G be a graph on n vertices, with complement Ḡ. 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 Ḡ is Ω(1/n). Moreover, if all degrees are Ω(n) and n-Ω(n), then the larger spectral gap of G and Ḡ is Θ(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 Ω(1/n). Finally, we provide a family of graphs such that the larger spectral gap of G and Ḡ is O(1/n3/4).