Spectra of lifted Ramanujan graphs

Eyal Lubetzky, Benny Sudakov, Van Vu

Research output: Contribution to journalArticlepeer-review


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.

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


  • Ramanujan graphs
  • Random lifts
  • Spectral expanders

ASJC Scopus subject areas

  • General Mathematics


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