### 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(ρ^{1/2}λ_{1}
^{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(Π^{2/3}λ_{1}
^{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≤d^{2/3-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 language | English (US) |
---|---|

Pages (from-to) | 1612-1645 |

Number of pages | 34 |

Journal | Advances in Mathematics |

Volume | 227 |

Issue number | 4 |

DOIs | |

State | Published - Jul 10 2011 |

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### Keywords

- Ramanujan graphs
- Random lifts
- Spectral expanders

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*227*(4), 1612-1645. https://doi.org/10.1016/j.aim.2011.03.016

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

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 227, no. 4, pp. 1612-1645. https://doi.org/10.1016/j.aim.2011.03.016

}

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 -