### Abstract

Consider a random regular graph with degree d and of size n. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortest-weight paths has aboutαlog n edges, where α is the unique solution of the equation α (d.2/d.1α) . α = d.3 d.2 for α > d.1/d.2.

Original language | English (US) |
---|---|

Pages (from-to) | 656-672 |

Number of pages | 17 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- First passage percolation
- Shortest-weight paths
- Weighted random graphs

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*28*(2), 656-672. https://doi.org/10.1137/120899534

**Shortest-Weight paths in random regular graphs.** / Amini, Leo Hamed; Peres, Yuval.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 28, no. 2, pp. 656-672. https://doi.org/10.1137/120899534

}

TY - JOUR

T1 - Shortest-Weight paths in random regular graphs

AU - Amini, Leo Hamed

AU - Peres, Yuval

PY - 2014

Y1 - 2014

N2 - Consider a random regular graph with degree d and of size n. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortest-weight paths has aboutαlog n edges, where α is the unique solution of the equation α (d.2/d.1α) . α = d.3 d.2 for α > d.1/d.2.

AB - Consider a random regular graph with degree d and of size n. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed d ≥ 3, we show that the longest of these shortest-weight paths has aboutαlog n edges, where α is the unique solution of the equation α (d.2/d.1α) . α = d.3 d.2 for α > d.1/d.2.

KW - First passage percolation

KW - Shortest-weight paths

KW - Weighted random graphs

UR - http://www.scopus.com/inward/record.url?scp=84904018647&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904018647&partnerID=8YFLogxK

U2 - 10.1137/120899534

DO - 10.1137/120899534

M3 - Article

AN - SCOPUS:84904018647

VL - 28

SP - 656

EP - 672

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -