Year: 2012
Type: preprint
Abstract: Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. 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 $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hatα\log n$ edges where $\hatα$ is the unique solution of the equation $α\log(\frac{d-2}{d-1}α) - α= \frac{d-3}{d-2}$, for $α> \frac{d-1}{d-2}$.
Source: arXiv (Cornell University)
Authors: Amini, Hamed, Peres, Yuval
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Austrian Science Fund
OpenAlex ID: W2951967911