### Abstract

Suppose each edge of the complete graph K_{n} is assigned a random weight chosen independently and uniformly from the unit interval [0,1]. A minimal spanning tree is a spanning tree of K_{n} with the minimum weight. It is easy to show that such a tree is unique almost surely. This paper concerns the number N_{n}(α) of vertices of degree α in the minimal spanning tree of K_{n}. For a positive integer α, Aldous (Random Struct. Algorithms 1 (1990) 383) proved that the expectation of N_{n}(α) is asymptotically γ(α)n, where γ(α) is a function of α given by explicit integrations. We develop an algorithm to generate the minimal spanning tree and Chernoff-type tail bound for N_{n}(α).

Original language | English |
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Pages (from-to) | 425-430 |

Number of pages | 6 |

Journal | Statistics and Probability Letters |

Volume | 64 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2003 Oct 1 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Statistics and Probability Letters*,

*64*(4), 425-430. https://doi.org/10.1016/S0167-7152(03)00208-6