@inproceedings{fc35993865b64c79b43f42b870e6a5b0,

title = "Ramanujan coverings of graphs",

abstract = "Let G be a finite connected graph, and let ρ be the spectral radius of its universal cover. For example, if G is k-regular then ρ = 2√k - 1. We show that for every r, there is an rcovering (a.k.a. an r-lift) of G where all the new eigenvalues are bounded from above by ρ. 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 Sr. 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 Marcus-Spielman-Srivastava (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 [-ρ,ρ].",

keywords = "Expander graphs, Graph coverings, Graph lifts, Interlacing polynomials, Matching polynomial, Ramanujan graphs",

author = "Chris Hall and Doron Puder and Sawin, {William F.}",

note = "Publisher Copyright: {\textcopyright} 2016 ACM.; 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 ; Conference date: 19-06-2016 Through 21-06-2016",

year = "2016",

month = jun,

day = "19",

doi = "10.1145/2897518.2897574",

language = "English (US)",

series = "Proceedings of the Annual ACM Symposium on Theory of Computing",

publisher = "Association for Computing Machinery",

pages = "533--541",

editor = "Yishay Mansour and Daniel Wichs",

booktitle = "STOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing",

}