[1505.08010] 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 we recently introduced.

View PDF HTML (experimental) 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…

saved by

• Didier Lampard

related reading

• [2408.01709] A spectral Lovász-Simonovits theoremarxiv.org
• Ramsey theory and strength of graphsarxiv.org
• probmethod_notes.pdfyufeizhao.com
• [2608.00140] Discrepancy Theory: An Algorithmic and Geometric Perspectivearxiv.org
• The Kadison-Singer Conjecture has beed Proved by Adam Marcus, Dan Spielman, and Nikhil Srivastavagilkalai.wordpress.com
• Research • Leonid Petrov. Integrable Probabilitylpetrov.cc
• Selector form of Weaver's conjecture, Feichtinger's conjecture, and frame sparsificationarxiv.org
• KS.pdfraghavsomani.github.io
• A simpler proof of the KPR theorem | tcs mathtcsmath.wordpress.com
• 1404.5236 Sum-of-Squares Proofs and the Quest toward Optimal Algorithmsarxiv.org
• Cover times - spectralarxiv.org
• Dynamical Approach to RMT.pdfpeople.math.harvard.edu