Ramanujan Coverings of Graphs

Let \(G\) be a finite connected graph, and let \(\rho\) be the spectral radius of its universal cover. For example, if \(G\) is \(k\)-regular then \(\rho=2\sqrt{k-1}\). We show that for every \(r\), there is an \(r\)-covering (a.k.a. an \(r\)-lift) of \(G\) where all the new eigenvalues are bounded from above by \(\rho\). It follows that a bipartite Ramanujan graph has a Ramanujan \(r\)-covering for every \(r\). This generalizes the \(r=2\) case due to Marcus, Spielman and Srivastava (2013). Every \(r\)-covering of \(G\) corresponds to a labeling of the edges of \(G\) by elements of the symmetric group \(S_{r}\). We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the \(r\)-th matching polynomial of \(G\) to be the average matching polynomial of all \(r\)-coverings of \(G\). We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside \(\left[-\rho,\rho\right]\).