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

In his Ph.D. thesis, Greenberg proved that if is the spectral radius of the universal cover of a finite graph X, then for each > 0, a positive proportion (depending only on and ) of the eigenvalues of X have absolute value at least . 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.