Relative Expanders or Weakly Relatively Ramanujan Graphs

Let G be a fixed graph with largest (adjacency matrix) eigenvalue λ0 and with its universal cover having spectral radius ρ. We show that a random cover of large degree over G has its “new” eigenvalues bounded in absolute value by roughly √ λ0ρ. 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 LubotzkyNagnibeda that showed that there is a tree all of whose finite 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. For the choice of G that specializes our theorem to the BroderShamir setting, our result slightly improves theirs. MSC 2000 numbers: Primary: 05C50; Secondary: 05C80, 68R10. ∗Departments of Computer Science and Mathematics, University of British Columbia, Vancouver, BC V6T 1Z4 (V6T 1Z2 for Mathematics), CANADA. [email protected]. Research supported in part by an NSERC grant.