@inproceedings{9184a8eb4a314c5786543045ec947fbe,

title = "Interlacing families I: Bipartite Ramanujan graphs of all degrees",

abstract = "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 b√ipartite graphs with all non-trivial eigenvalues bounded by c - 1 + √ d - 1, for all c, d ≥ 3. Our proof exploits a new technique for demonstrating the existence of useful combinatorial objects that we call the {"}method of interlacing polynomials{"}.",

keywords = "Lifts of graphs, Matching polynomial, Ramanujan graph",

author = "Adam Marcus and Spielman, {Daniel A.} and Nikhil Srivastava",

year = "2013",

doi = "10.1109/FOCS.2013.63",

language = "English (US)",

isbn = "9780769551357",

series = "Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS",

pages = "529--537",

booktitle = "Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013",

note = "2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 ; Conference date: 27-10-2013 Through 29-10-2013",

}