COOKIES: By using this website you agree that we can place Google Analytics Cookies on your device for performance monitoring. |

University of Cambridge > Talks.cam > Combinatorics Seminar > Ramanujan Graphs and Finite Free Convolutions of Polynomials

## Ramanujan Graphs and Finite Free Convolutions of PolynomialsAdd to your list(s) Download to your calendar using vCal - Dan Spielman (Yale University)
- Tuesday 02 June 2015, 14:30-15:30
- MR14.
If you have a question about this talk, please contact Andrew Thomason. We use the method of interlacing polynomials and a finite dimensional approach to free probability to prove the existence of bipartite Ramanujan graphs of every degree and number of vertices. No prior knowledge of Ramanujan graphs or free probability will be assumed. Ramanujan graphs are defined in terms of the eigenvalues of their adjacency or Laplacian matrices. In this spectral perspective, they are the best possible expanders. Infinite families of Ramanujan graphs were first shown to exist by Margulis and Lubotzky, Phillips and Sarnak using Deligne’s proof of the Ramanujan conjecture. These constructions were sporadic, only producing graphs of special degrees and numbers of vertices. In this talk, we outline an elementary proof of the existence of bipartite Ramanujan graphs of very degree and number of vertices. We do this by considering the expected characteristic polynomial of a random d-regular graph. We develop finite analogs of results in free probability to compute this polynomial and to bound its roots. By proving that this polynomial is the average of polynomials in an interlacing family, we then prove there exists a graph in the distribution whose eigenvalues satisfy the Ramanujan bound. These results are joint work with Adam Marcus and Nikhil Srivastava. This talk is part of the Combinatorics Seminar series. ## This talk is included in these lists:- All CMS events
- All Talks (aka the CURE list)
- CMS Events
- Combinatorics Seminar
- DPMMS Lists
- DPMMS Pure Maths Seminar
- DPMMS info aggregator
- DPMMS lists
- MR14
- School of Physical Sciences
Note that ex-directory lists are not shown. |
## Other listsSt Johns Women Society Naked mole-rats Modelling in Diabetes## Other talksCharacterizing Immunogenetic Mechanisms through HIV Disease Analysis Develop a tool for inferring symptoms from prescriptions histories for cancer patients Ethics for the working mathematician, seminar 11: Our mathematical identity. Acceleration of tropical cyclogenesis by self-aggregation feedbacks Symbolic AI in Computational Biology; applications to disease gene and drug target identification The formation of high density dust rings and clumps: the role of vorticity |