### Abstract

Let d >= 3 be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random d-regular graph with n vertices. (The asymptotics are as n -> infinity, restricted to even n if d is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) d. Numerical evidence is presented which supports our conjecture.

Original language | English |
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Article number | P1.45 |

Journal | The Electronic Journal of Combinatorics |

Volume | 21 |

Issue number | 1 |

Number of pages | 26 |

ISSN | 1097-1440 |

Publication status | Published - 2014 |

### Keywords

- Spanning trees
- Random regular graphs
- Small subgraph conditioning

## Cite this

Greenhill, C., Kwan, M., & Wind, D. K. (2014). On the number of spanning trees in random regular graphs.

*The Electronic Journal of Combinatorics*,*21*(1), [P1.45]. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p45