Explicit near-Ramanujan graphs of every degree

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