The spectrum of random lifts
Abstract
For a fixed dregular graph H, a random nlift is obtained by replacing each vertex v of H by a "fibre" containing n vertices, then placing a uniformly random matching between fibres corresponding to adjacent vertices of H. We show that with extremely high probability, all eigenvalues of the lift that are not eigenvalues of H, have order O(sqrt(d)). In particular, if H is Ramanujan then its nlift is with high probability nearly Ramanujan. We also show that any exceptionally large eigenvalues of the nlift that are not eigenvalues of H, are overwhelmingly likely to have been caused by a dense subgraph of size O(E(H)).
 Publication:

arXiv eprints
 Pub Date:
 December 2010
 arXiv:
 arXiv:1012.4097
 Bibcode:
 2010arXiv1012.4097A
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability;
 05C80;
 05C50;
 06C05
 EPrint:
 35 pages