TY - JOUR

T1 - Spectra of lifted Ramanujan graphs

AU - Lubetzky, Eyal

AU - Sudakov, Benny

AU - Vu, Van

N1 - Funding Information:
* Corresponding author. E-mail addresses: eyal@microsoft.com (E. Lubetzky), bsudakov@math.ucla.edu (B. Sudakov), vanvu@math.rutgers.edu (V. Vu). 1 The author is supported by NSF CAREER award 0812005 and a USA-Israeli BSF grant. 2 The author is supported by research grants DMS-0901216 and AFOSAR-FA-9550-09-1-0167.

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(ρ1/2λ11/2) 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(Π2/3λ11/3). 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(ρ1/2λ11/2) 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(Π2/3λ11/3). 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

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U2 - 10.1016/j.aim.2011.03.016

DO - 10.1016/j.aim.2011.03.016

M3 - Article

AN - SCOPUS:79956050252

VL - 227

SP - 1612

EP - 1645

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -