An Alon-Boppana--type bound for very dense graphs, with applications to max-cut

For any $\epsilon > 0$, we show that if $G$ is a regular graph on $n \gg_\epsilon 1$ vertices that is $\epsilon$-far (differs by at least $\epsilon n^2$ edges) from any Tur\'{a}n graph, then its second eigenvalue $\lambda_2$ satisfies $$\lambda_2 \geq n^{1/4 - \epsilon}.$$ The exponent $1/4$ is optimal. Our result generalizes an analogous bound, independently obtained by Balla, R\"{a}ty -- Sudakov-Tomon, and Ihringer, which only applies to graphs with density at most $\frac{1}{2}$. Up to a lower-order factor, this confirms a conjecture of R\"{a}ty, Sudakov and Tomon. Our spectral approach has interesting applications to max-cut. First, we show that if a graph $G$, on $n \gg_\epsilon 1$ vertices and $m$ edges, is $\epsilon$-far from a disjoint union of cliques, then it has a max-cut of size at least $$\frac{m}{2} + n^{1.01}.$$ Our result improves upon a classical result of Edwards by a non-trivial polynomial factor, making progress towards another conjecture of R\"{a}ty, Sudakov and Tomon. As another application of our method, we show that if a graph $G$ is $H$-free and has $m$ edges, then $G$ has a max-cut of size at least $$\frac{m}{2} + c_H m^{0.5001}$$ where $c_H > 0$ is some constant depending on $H$ only. This result makes progress towards a conjecture of Alon, Bollob\'{a}s, Krivelevich and Sudakov, and answers recent questions by Glock-Janzer-Sudakov and Balla-Janzer-Sudakov.