Ramanujan coverings of graphs

Chris Hall, Doron Puder, William F. Sawin

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

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].

Original languageEnglish
Title of host publicationSTOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing
EditorsYishay Mansour, Daniel Wichs
PublisherAssociation for Computing Machinery
Pages533-541
Number of pages9
ISBN (Electronic)9781450341325
DOIs
StatePublished - 19 Jun 2016
Externally publishedYes
Event48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 - Cambridge, United States
Duration: 19 Jun 201621 Jun 2016

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
Volume19-21-June-2016
ISSN (Print)0737-8017

Conference

Conference48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016
Country/TerritoryUnited States
CityCambridge
Period19/06/1621/06/16

Funding

FundersFunder number
IAS NSFDMS-1128155
National Science FoundationDGE-1148900
Simons Foundation245619

    Keywords

    • Expander graphs
    • Graph coverings
    • Graph lifts
    • Interlacing polynomials
    • Matching polynomial
    • Ramanujan graphs

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