Advances in Applied Probability
- Adv. in Appl. Probab.
- Volume 48, Number 3 (2016), 848-864.
On the relation between graph distance and Euclidean distance in random geometric graphs
Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 848-864.
First available in Project Euclid: 19 September 2016
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35]
Díaz, J.; Mitsche, D.; Perarnau, G.; Pérez-Giménez, X. On the relation between graph distance and Euclidean distance in random geometric graphs. Adv. in Appl. Probab. 48 (2016), no. 3, 848--864. https://projecteuclid.org/euclid.aap/1474296318