Explicit Expanders of Every Degree and Size

Noga Alon*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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

Original languageEnglish
Pages (from-to)447-463
Number of pages17
Issue number4
StatePublished - Aug 2021


FundersFunder number
National Science FoundationDMS-1855464
Israel Science Foundation281/17
Bonfils-Stanton Foundation2018267
Simons Foundation


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