Interlacing families I: Bipartite Ramanujan graphs of all degrees

• 成果类型: Article
• 署名作者: Marcus, Adam W.; Spielman, Daniel A.; Srivastava, Nikhil
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2015.182.1.7
• 发表日期: 2015
• 页码: 307-325
• 关键词: random lifts; stable polynomials; SPECTRA; INDEPENDENCE; THEOREM

摘要

We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also establish the existence of infinite families of irregular Ramanujan graphs, whose eigenvalues are bounded by the spectral radius of their universal cover. Such families were conjectured to exist by Linial and others. In particular, we prove the existence of infinite families of (c, d)-biregular bipartite graphs with all nontrivial eigenvalues bounded by root c-1 + root d-1 for all c,d >= 3. Our proof exploits a new technique for controlling the eigenvalues of certain random matrices, which we call the method of interlacing polynomials.