On spectral radii of unraveled balls

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 root d-1 cos(pi/r+1). (C) 2018 Elsevier Inc. All rights reserved.