Explicit Expanders of Every Degree and Size

An (n, d, ?)-graph is a d regular graph on n vertices in which the absolute value of any nontrivial eigenvalue at most ?. For constant ? 3, ? > 0 and all sufficiently large we show that there deterministic poly(n) time algorithm outputs an (on exactly vertices) with $$\lambda \le 2\sqrt {d - 1} + $$ . d=p 2 p ? 1 mod 4 prime n, describe strongly explicit construction \sqrt {2\left({d \right)} 2} o\left(1 \right)\left({< \left({1 \right)\sqrt \right)$$ , o(1) term tending to as tends infinity. every ?> 0, d> d0(?) n>n0(d, ?) present (m, $$\lambda<\left({2 +} m = o(n). All constructions are obtained by starting known Ramanujan or nearly graphs modifying packing them appropriate way. The spectral analysis relies delocalization eigenvectors cycle-free neighborhoods.