On the relation between graph distance and Euclidean distance in random geometric graphs

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(√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 d_E(u, v) conditional on d_E(u, v).