### Abstract

Let d be a (large) integer. Given n≥2d, let A_{n} be the adjacency matrix of a random directed d-regular graph on n vertices, with the uniform distribution. We show that the rank of A_{n} is at least n−1 with probability going to one as n grows to infinity. The proof combines the well known method of simple switchings and a recent result of the authors on delocalization of eigenvectors of A_{n}.

Original language | English (US) |
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Pages (from-to) | 103-110 |

Number of pages | 8 |

Journal | Journal of Complexity |

Volume | 48 |

DOIs | |

State | Published - Oct 2018 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Mathematics(all)
- Control and Optimization
- Applied Mathematics

### Keywords

- Random matrices
- Random regular graphs
- Rank
- Singularity probability

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## Cite this

Litvak, A. E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N., & Youssef, P. (2018). The rank of random regular digraphs of constant degree.

*Journal of Complexity*,*48*, 103-110. https://doi.org/10.1016/j.jco.2018.05.004