Relative Expanders or Weakly Relatively

Let G be a xed graph with largest (adjacency matrix) eigenvalue 0 and with its universal cover having spectral radius .W e show that a random cover of large degree over G has its \new" eigenvalues bounded in absolute value by roughly p 0. This gives a positive result about nite quotients of certain trees having \small" eigenvalues, provided we ignore the \old" eigenvalues. This positive result contrasts with the negative result of LubotzkyNagnibeda that showed that there is a tree all of whose nite quotients are not \Ramanujan" in the sense of Lubotzky-Philips-Sarnak and 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 .F or the choice ofG that specializes our theorem to the BroderShamir setting, our result slightly improves theirs.