It is shown that if a $d$-regular graph contains $s$ vertices so that the distance between any pair is at least $4k$, then its adjacency matrix has at least $s$ eigenvalues which are at least $2 d-1 \big(\pi2 k\big)$. A similar result has been proved by Friedman using more sophisticated tools.
Description
Tight Estimates for Eigenvalues of Regular Graphs | Nilli | The Electronic Journal of Combinatorics
%0 Journal Article
%1 nilli12
%A Nilli, A.
%D 2012
%K alon-boppana eigenvalues expander graph.theory ramanujan serre
%T Tight Estimates for Eigenvalues of Regular Graphs
%U http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1n9
%X It is shown that if a $d$-regular graph contains $s$ vertices so that the distance between any pair is at least $4k$, then its adjacency matrix has at least $s$ eigenvalues which are at least $2 d-1 \big(\pi2 k\big)$. A similar result has been proved by Friedman using more sophisticated tools.
@article{nilli12,
abstract = { It is shown that if a $d$-regular graph contains $s$ vertices so that the distance between any pair is at least $4k$, then its adjacency matrix has at least $s$ eigenvalues which are at least $2 \sqrt {d-1} \cos \big({\pi\over 2 k}\big)$. A similar result has been proved by Friedman using more sophisticated tools. },
added-at = {2017-02-24T12:52:38.000+0100},
author = {Nilli, A.},
biburl = {https://www.bibsonomy.org/bibtex/238a8f58e0b51e90cd506f42a9fd52c75/ytyoun},
description = {Tight Estimates for Eigenvalues of Regular Graphs | Nilli | The Electronic Journal of Combinatorics},
interhash = {c10a26d3370a97cc64b4b4e53d00377c},
intrahash = {38a8f58e0b51e90cd506f42a9fd52c75},
keywords = {alon-boppana eigenvalues expander graph.theory ramanujan serre},
timestamp = {2017-02-24T12:52:38.000+0100},
title = {Tight Estimates for Eigenvalues of Regular Graphs},
url = {http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1n9},
year = 2012
}