Spectra of lifted Ramanujan graphs

Eyal Lubetzky, Benny Sudakov, Van Vu

Research output: Contribution to journalArticle

Abstract

A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection Π, and H has all edges of the form (i,u),(Π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G be a graph with largest eigenvalue λ1 and let ρ be the spectral radius of its universal cover. Friedman (2003) [12] proved that every "new" eigenvalue of a random lift of G is O(ρ1/2λ1 1/2) with high probability, and conjectured a bound of Π+o(1), which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved FriedmanΠs bound to O(Π2/3λ1 1/3). For d-regular graphs, where ρ1=d and d-1, this translates to a bound of O(d2/3), compared to the conjectured 2√d-1. Here we analyze the spectrum of a random n-lift of a d-regular graph whose nontrivial eigenvalues are all at most λ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is O((λVρ)logρ). This result is tight up to a logarithmic factor, and for λ≤d2/3-ε it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical n-lift of a Ramanujan graph is nearly Ramanujan.

Original languageEnglish (US)
Pages (from-to)1612-1645
Number of pages34
JournalAdvances in Mathematics
Volume227
Issue number4
DOIs
StatePublished - Jul 10 2011

Fingerprint

Ramanujan Graphs
Ramanujan
Graph in graph theory
Regular Graph
Eigenvalue
Absolute value
Cover
Universal Cover
Largest Eigenvalue
Spectral Radius
Bijection
Logarithmic
Upper bound
Imply

Keywords

  • Ramanujan graphs
  • Random lifts
  • Spectral expanders

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Spectra of lifted Ramanujan graphs. / Lubetzky, Eyal; Sudakov, Benny; Vu, Van.

In: Advances in Mathematics, Vol. 227, No. 4, 10.07.2011, p. 1612-1645.

Research output: Contribution to journalArticle

Lubetzky, Eyal ; Sudakov, Benny ; Vu, Van. / Spectra of lifted Ramanujan graphs. In: Advances in Mathematics. 2011 ; Vol. 227, No. 4. pp. 1612-1645.
@article{3a44a6b11110472ea5ce36adce79e2d7,
title = "Spectra of lifted Ramanujan graphs",
abstract = "A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection Π, and H has all edges of the form (i,u),(Π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G be a graph with largest eigenvalue λ1 and let ρ be the spectral radius of its universal cover. Friedman (2003) [12] proved that every {"}new{"} eigenvalue of a random lift of G is O(ρ1/2λ1 1/2) with high probability, and conjectured a bound of Π+o(1), which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved FriedmanΠs bound to O(Π2/3λ1 1/3). For d-regular graphs, where ρ1=d and d-1, this translates to a bound of O(d2/3), compared to the conjectured 2√d-1. Here we analyze the spectrum of a random n-lift of a d-regular graph whose nontrivial eigenvalues are all at most λ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is O((λVρ)logρ). This result is tight up to a logarithmic factor, and for λ≤d2/3-ε it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical n-lift of a Ramanujan graph is nearly Ramanujan.",
keywords = "Ramanujan graphs, Random lifts, Spectral expanders",
author = "Eyal Lubetzky and Benny Sudakov and Van Vu",
year = "2011",
month = "7",
day = "10",
doi = "10.1016/j.aim.2011.03.016",
language = "English (US)",
volume = "227",
pages = "1612--1645",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",
number = "4",

}

TY - JOUR

T1 - Spectra of lifted Ramanujan graphs

AU - Lubetzky, Eyal

AU - Sudakov, Benny

AU - Vu, Van

PY - 2011/7/10

Y1 - 2011/7/10

N2 - A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection Π, and H has all edges of the form (i,u),(Π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G be a graph with largest eigenvalue λ1 and let ρ be the spectral radius of its universal cover. Friedman (2003) [12] proved that every "new" eigenvalue of a random lift of G is O(ρ1/2λ1 1/2) with high probability, and conjectured a bound of Π+o(1), which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved FriedmanΠs bound to O(Π2/3λ1 1/3). For d-regular graphs, where ρ1=d and d-1, this translates to a bound of O(d2/3), compared to the conjectured 2√d-1. Here we analyze the spectrum of a random n-lift of a d-regular graph whose nontrivial eigenvalues are all at most λ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is O((λVρ)logρ). This result is tight up to a logarithmic factor, and for λ≤d2/3-ε it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical n-lift of a Ramanujan graph is nearly Ramanujan.

AB - A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection Π, and H has all edges of the form (i,u),(Π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G be a graph with largest eigenvalue λ1 and let ρ be the spectral radius of its universal cover. Friedman (2003) [12] proved that every "new" eigenvalue of a random lift of G is O(ρ1/2λ1 1/2) with high probability, and conjectured a bound of Π+o(1), which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved FriedmanΠs bound to O(Π2/3λ1 1/3). For d-regular graphs, where ρ1=d and d-1, this translates to a bound of O(d2/3), compared to the conjectured 2√d-1. Here we analyze the spectrum of a random n-lift of a d-regular graph whose nontrivial eigenvalues are all at most λ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is O((λVρ)logρ). This result is tight up to a logarithmic factor, and for λ≤d2/3-ε it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical n-lift of a Ramanujan graph is nearly Ramanujan.

KW - Ramanujan graphs

KW - Random lifts

KW - Spectral expanders

UR - http://www.scopus.com/inward/record.url?scp=79956050252&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79956050252&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2011.03.016

DO - 10.1016/j.aim.2011.03.016

M3 - Article

VL - 227

SP - 1612

EP - 1645

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -