On the Expected Maximum Degree of Gabriel and Yao Graphs

TL;DR

This paper analyzes the maximum degree of Gabriel and Yao graphs formed from uniformly random points in a unit square, showing it grows logarithmically with a specific rate, which is important for understanding network properties.

Contribution

It establishes the asymptotic growth rate of the maximum degree in Gabriel and Yao graphs for random point sets, providing new theoretical insights.

Findings

Maximum degree grows as Θ(log n / log log n)

Results are probabilistic and hold with high probability

Implications for wireless network design

Abstract

Motivated by applications of Gabriel graphs and Yao graphs in wireless ad-hoc networks, we show that the maximal degree of a random Gabriel graph or Yao graph defined on $n$ points drawn uniformly at random from a unit square grows as $\Theta (\log n / \log \log n)$ in probability.