On spectral radii of unraveled balls
Abstract
Given a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We prove a lower bound on the maximum spectral radius of unraveled balls of fixed radius, and we show, among other things, that if the average degree of $G$ after deleting any ball of radius $r$ is at least $d$ then its second largest eigenvalue is at least $2\sqrt{d1}\cos(\frac{\pi}{r+1})$.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.06719
 Bibcode:
 2017arXiv171006719J
 Keywords:

 Mathematics  Combinatorics;
 05C50;
 60J10;
 15A42
 EPrint:
 8 pages, accepted to J. Comb. Theory B, corrections suggested by the referees have been incorporated