@inproceedings{d0c9388e8c024bb0ac8ece0c11e3ae1c,

title = "Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes",

abstract = "We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions introduced recently by the authors.",

keywords = "Ramanujan graphs, expected characteristic polynomials, finite free probability",

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

note = "Funding Information: This research was partially supported by NSF grant CCF-1111257, an NSF Mathematical Sciences Postdoctoral Research Fellowship, Grant No. DMS-0902962, a Simons Investigator Award to Daniel Spielman, and a MacArthur Fellowship.; 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 ; Conference date: 17-10-2015 Through 20-10-2015",

year = "2015",

month = dec,

day = "11",

doi = "10.1109/FOCS.2015.87",

language = "English (US)",

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

publisher = "IEEE Computer Society",

pages = "1358--1377",

booktitle = "Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015",

address = "United States",

}