Small Lifts of Expander Graphs are Expanding

A k-lift of an n-vertex base-graph G is a graph H on n× k vertices, where each vertex of G is replaced by k vertices and each edge (u, v) in G is replaced by a matching representing a bijection πuv so that the edges of H are of the form ( (u, i), (v, πuv(i)) ) . H is a (uniformly) random lift of G if for every edge (u, v) the bijection πuv is chosen uniformly and independently at random. The main motivation for studying lifts has been understanding Ramanujan expander graphs via two key questions: Is a “typical” lift of an expander graph also an expander; and how can we (efficiently) construct Ramanujan expanders using lifts? Lately, there has been an increased interest in lifts and their close relation to the notorious Unique Games Conjecture [Kho02]. In this paper, we analyze the spectrum of random k-lifts of d-regular graphs. We show that, for random shift k-lifts, if all the nontrivial eigenvalues of the base graph G are at most λ in absolute value, then with high probability depending only on the number n of nodes of G (and not on k), the absolute value of every nontrivial eigenvalue of the lift is at most O(λ). Moreover, if G is moderately expanding, then this bound can be improved to λ +O( √ d). While previous results on random lifts were asymptotically true with high probability in the degree of the lift k, our result is the first upperbound on spectra of lifts for bounded k. In particular, it implies that a typical small lift of a Ramanujan graph is almost Ramanujan, and we believe it will prove crucial in constructing large Ramanujan expanders of all degrees. We also establish a novel characterization of the spectrum of shift lifts by the spectrum of certain k symmetric matrices, that generalize the signed adjacency matrix (e.g. see [BL06]). We believe this characterization is of independent interest. ∗[email protected], University of Illinois Urbana-Champaign †[email protected], University of Illinois Urbana-Champaign ‡[email protected], University of Illinois Urbana-Champaign