### 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) |
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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 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Amini, H., & Peres, Y. (2014). Shortest-Weight paths in random regular graphs.

*SIAM Journal on Discrete Mathematics*,*28*(2), 656-672. https://doi.org/10.1137/120899534