Explicit near-Ramanujan graphs of every degree

TL;DR

This paper presents a deterministic polynomial-time algorithm for constructing explicit near-Ramanujan graphs of any fixed degree, with eigenvalues close to the optimal spectral bound, for all sufficiently large graphs.

Contribution

It provides the first explicit, deterministic construction of near-Ramanujan graphs for every degree $d \\geq 3$, improving upon previous probabilistic methods.

Findings

Constructs explicit near-Ramanujan graphs for all degrees $d \\geq 3$

Achieves eigenvalue bounds within epsilon of the Ramanujan bound

Operates in polynomial time for graphs of size proportional to $n$

Abstract

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 $2\sqrt{d-1} + \epsilon$ (excluding the single trivial eigenvalue of~$d$).