%0 Conference Paper %1 marcus13 %A Marcus, Adam %A Spielman, Daniel A. %A Srivastava, Nikhil %B Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on %D 2013 %K alon-boppana bipartite characteristic expander graph.theory interlacing lift polynomial ramanujan real-rooted root root-free sparsification stable %P 529-537 %R 10.1109/FOCS.2013.63 %T Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees %X 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 $c-1+sqrtd-1$, 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".

@inproceedings{marcus13, 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 bipartite graphs with all non-trivial eigenvalues bounded by $\sqrt{c-1}+sqrt{d-1}$, for all $c, d \geq 3$. Our proof exploits a new technique for demonstrating the existence of useful combinatorial objects that we call the "method of interlacing polynomials".}, added-at = {2014-03-13T15:30:00.000+0100}, archiveprefix = {arXiv}, author = {Marcus, Adam and Spielman, Daniel A. and Srivastava, Nikhil}, biburl = {https://www.bibsonomy.org/bibtex/202369cce47ba498ee732991fefdb86c4/ytyoun}, booktitle = {Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on}, doi = {10.1109/FOCS.2013.63}, eprint = {1304.4132}, interhash = {8e7054ad8d35e9059c810063b0bcbb97}, intrahash = {02369cce47ba498ee732991fefdb86c4}, issn = {0272-5428}, keywords = {alon-boppana bipartite characteristic expander graph.theory interlacing lift polynomial ramanujan real-rooted root root-free sparsification stable}, month = oct, pages = {529-537}, primaryclass = {math.CO}, timestamp = {2017-03-17T10:28:17.000+0100}, title = {{Interlacing Families I}: Bipartite {Ramanujan} Graphs of All Degrees}, year = 2013 }