## 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

J. Díaz, D. Mitsche, G. Perarnau, and X. Pérez-Giménez

#### Abstract

Given any two vertices *u*, *v* of a random geometric graph G(*n*, *r*), denote by *d*_{E}(*u*, *v*) their Euclidean distance and by *d*_{E}(*u*, *v*) their graph distance. The problem of finding upper bounds on *d*_{G}(*u*, *v*) conditional on *d*_{E}(*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(√log*n*) (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 *d*_{E}(*u*, *v*) conditional on *d*_{E}(*u*, *v*).

#### Article information

**Source**

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 848-864.

**Dates**

First available in Project Euclid: 19 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1474296318

**Mathematical Reviews number (MathSciNet)**

MR3568895

**Zentralblatt MATH identifier**

1348.05188

**Subjects**

Primary: 05C80: Random graphs [See also 60B20]

Secondary: 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35]

**Keywords**

Random geometric graph graph distance Euclidean distance diameter

#### Citation

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