Let $G$ be a fixed graph with largest (adjacency matrix) eigenvalue
$\lambda\sb 0$ and with its universal cover having spectral radius
$rho$. We show that a random cover of large degree over $G$ has its
"new" eigenvalues bounded in absolute value by roughly
$\sqrt{\lambda\sb 0\rho}$.
¶ This gives a positive result about finite quotients of certain
trees having "small" eigenvalues, provided we ignore the "old"
eigenvalues. This positive result contrasts with the negative result
of A. Lubotzky and T. Nagnibeda which showed that there is a tree all
of whose finite quotients are not "Ramanujan" in the sense of
Lubotzky, R. Phillips, and P. Sarnakand of Y. Greenberg.
¶ Our main result is a "relative version" of the Broder-Shamir bound
on eigenvalues of random regular graphs. Some of their combinatorial
techniques are replaced by spectral techniques on the universal cover
of $G$. For the choice of $G$ that specializes our main theorem to the
Broder-Shamir setting, our result slightly improves theirs.