The length of the shortest edge of a graph on a sphere
type: Item
instance of: scholarly article
title: The length of the shortest edge of a graph on a sphere
zbMATH Open document ID: 1014.05036
author: H. Maehara
published in: European Journal of Combinatorics
publication date: 2003-03-18T00:00:00Z
review text: Let \(S^{d}\) denote a unit sphere in the \((d+1)\)-dimensional Euclidean space \(\mathbb{R}^{d+1}\) \((d\geq 1)\). For a simple graph \(G_{E}\) with edge set \(E\), take independent random points \(x_{k}\) \((k\in V(G_{E}))\) on \(S^{d}\), and let \(D_{E}\) be the minimum value of the spherical distance between \(x_{i},x_{j}\) for \(\{i,j\}\in E\). Then \(D_{E}\) is a random variable. The main result of this paper is that \(|E|D_{E}^{d}\) is asymptotically distributed (as \(|E|\rightarrow \infty)\) according to the exponential distribution with mean \(dB(\frac{1}{2},\frac{d}{2})\), where \(B(p,q)\) is the beta function.
Mathematics Subject Classification ID: 05C35
zbMATH DE Number: 1884157
zbMATH Keywords: Euclidean space
zbMATH Keywords: spherical distance
zbMATH Keywords: random point
zbMATH Keywords: asymptotic distribution
zbMATH Keywords: beta function
reviewed by: Ioan Tomescu
MaRDI profile type: Publication
full work available at URL: eujc.2002.0598
OpenAlex ID: W2065534605
cites work: Some geometric applications of the beta distribution
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cites work: TESTING THE LARGEST OF A SET OF CORRELATION COEFFICIENTS
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DOI: 10.1006/EUJC.2002.0598
description: scientific article; zbMATH DE number 1884157