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

In his Ph.D. thesis, Greenberg proved that if p(5) is the spectral radius of the universal cover 5 of a finite graph X, then for each epsilon > 0, a positive proportion (depending only on (X) over tilde and epsilon) of the eigenvalues of X have absolute value at least p ((X) over tilde) - epsilon. 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. (c) 2006 Elsevier Inc. All rights reserved.