### 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 b√ipartite graphs with all non-trivial eigenvalues bounded by c - 1 + √ d - 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".

Original language | English (US) |
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Title of host publication | Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 |

Pages | 529-537 |

Number of pages | 9 |

DOIs | |

State | Published - Dec 1 2013 |

Externally published | Yes |

Event | 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 - Berkeley, CA, United States Duration: Oct 27 2013 → Oct 29 2013 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 |
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Country | United States |

City | Berkeley, CA |

Period | 10/27/13 → 10/29/13 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

### Keywords

- Lifts of graphs
- Matching polynomial
- Ramanujan graph

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## Cite this

*Proceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013*(pp. 529-537). [6686189] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2013.63