Gaps Between Almost-Primes and a Construction of Almost-Ramanujan Graphs

TL;DR

This paper presents an explicit construction of infinite families of k-regular graphs with second largest eigenvalues bounded by O(k^{1/2}), solving a longstanding open problem in graph theory.

Contribution

It provides a novel explicit method to construct almost-Ramanujan graphs for all degrees k ≥ 3, advancing understanding of spectral properties of regular graphs.

Findings

Constructed infinite families of k-regular graphs with optimal spectral bounds

Resolved an open problem by Reingold, Vadhan, and Wigderson

Established explicit constructions for almost-Ramanujan graphs

Abstract

For all $k \geq 3$, we show how one can explicitly construct an infinite family of $k$-regular graphs all of which have second largest eigenvalue satisfying the bound $O(k^{1/2})$. This resolves an open problem of Reingold, Vadhan and Wigderson.