2013/04/15 by Adam W. Marcus, Daniel A. Spielman, Marcus, Adam +3 · 16 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1304.4132
openalex publication_date 2013/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 non-trivial eigenvalues bounded by sqrtc-1+sqrtd-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'".