Explicit expanders of every degree and size

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