### 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 introduced recently by the authors.

Original language | English (US) |
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Title of host publication | Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015 |

Publisher | IEEE Computer Society |

Pages | 1358-1377 |

Number of pages | 20 |

ISBN (Electronic) | 9781467381918 |

DOIs | |

State | Published - Dec 11 2015 |

Event | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States Duration: Oct 17 2015 → Oct 20 2015 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2015-December |

ISSN (Print) | 0272-5428 |

### Other

Other | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 |
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Country | United States |

City | Berkeley |

Period | 10/17/15 → 10/20/15 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

### Keywords

- Ramanujan graphs
- expected characteristic polynomials
- finite free probability

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## Cite this

*Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015*(pp. 1358-1377). [7354461] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2015-December). IEEE Computer Society. https://doi.org/10.1109/FOCS.2015.87