TY - JOUR

T1 - Explicit Expanders of Every Degree and Size

AU - Alon, Noga

N1 - Publisher Copyright:
© 2021, János Bolyai Mathematical Society and Springer-Verlag.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

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U2 - 10.1007/s00493-020-4429-x

DO - 10.1007/s00493-020-4429-x

M3 - Article

AN - SCOPUS:85100277678

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

ER -