2015/05/29 by Adam W. Marcus, Nikhil Srivastava, Marcus, Adam W. +3 · 2 citations
Mathematics · #05C50 #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1505.08010
openalex publication_date 2015/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.