In 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.
%0 Journal Article
%1 cioaba06
%A Cioaba, Sebastian M.
%D 2006
%J Linear Algebra and its Applications
%K alon-boppana eigenvalues graph.theory ramanujan serre spectral_graph_theory
%N 2
%P 776--782
%R 10.1016/j.laa.2005.12.020
%T Eigenvalues of Graphs and a Simple Proof of a Theorem of Greenberg
%V 416
%X In 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.
@article{cioaba06,
abstract = {In 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.},
added-at = {2016-10-29T16:36:07.000+0200},
author = {Cioab\u{a}, Sebastian M.},
biburl = {https://www.bibsonomy.org/bibtex/206356dc9eda1ae36f8a9275de93db99a/ytyoun},
doi = {10.1016/j.laa.2005.12.020},
interhash = {93350297c67e9ff17ae460fdc6918f53},
intrahash = {06356dc9eda1ae36f8a9275de93db99a},
issn = {0024-3795},
journal = {Linear Algebra and its Applications},
keywords = {alon-boppana eigenvalues graph.theory ramanujan serre spectral_graph_theory},
number = 2,
pages = {776--782},
timestamp = {2017-02-24T09:49:27.000+0100},
title = {Eigenvalues of Graphs and a Simple Proof of a Theorem of {Greenberg}},
volume = 416,
year = 2006
}