  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 $2\sqrt{d-1}\cos(\frac{\pi}{r+1})$.
