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 λ 1 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 ρ + o ( 1 ) , which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved Friedmanʼs bound to O ( ρ 2 / 3 λ 1 1 / 3 ) . For d-regular graphs, where λ 1 = d and ρ = 2 d − 1 , this translates to a bound of O ( d 2 / 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 ( ( λ ∨ ρ ) log ρ ) . 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.