Eigenvalues of graphs and a simple proof of a theorem of Greenberg

AbstractIn his Ph.D. thesis, Greenberg proved that if ρ(X∼) is the spectral radius of the universal cover X∼ of a finite graph X, then for each ϵ>0, a positive proportion (depending only on X∼ and ϵ) of the eigenvalues of X have absolute value at least ρ(X∼)-ϵ. In this paper, we show that the same result holds true if we remove absolute from the previous result. We also prove an analogue result for the smallest eigenvalues of X