# Shortest-Weight paths in random regular graphs

Leo Hamed Amini, Yuval Peres

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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) 656-672 17 SIAM Journal on Discrete Mathematics 28 2 https://doi.org/10.1137/120899534 Published - 2014 Yes

### Fingerprint

Regular Graph
Random Graphs
Precise Asymptotics
Path
Identically distributed
Unique Solution
Assign
Random variable
Vertex of a graph

### Keywords

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

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

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

In: SIAM Journal on Discrete Mathematics, Vol. 28, No. 2, 2014, p. 656-672.

Research output: Contribution to journalArticle

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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.

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