Deterministic nonbipartite Ramanujan graphs in every fixed degree

For every fixed integer d ≥ 3, we give a deterministic algorithm that constructs a simple nonbipartite d-regular Ramanujan graph on every sufficiently large even number n of vertices. It outputs the full adjacency list in polynomially many bit operations, with an exponent that may depend on d. Every nonconstant adjacency eigenvalue lies strictly between $-2\sqrt{d-1}$ and $2\sqrt{d-1}$.