## Abstract

We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigidity result. If most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. In particular, G contains few short cycles. In contrast, we show that d-regular unimodular random graphs with maximal growth are not necessarily trees.

Original language | English |
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Pages (from-to) | 1601-1646 |

Number of pages | 46 |

Journal | Annals of Probability |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2016 |

## Keywords

- Girth
- Mass transport principal
- Ramanujan graphs
- Spectral radius
- Unimodular random graphs