Explicit near-Ramanujan graphs of every degree

For every constant $d geq 3$ and $epsilon > 0$, we give a deterministic $mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $Theta(n)$ vertices that is $epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2sqrt{d-1} + epsilon$ (excluding the single trivial eigenvalue of~$d$).