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

Given any two vertices u, v of a random geometric graph, denote by dE(u, v) their Euclidean distance and by dG(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) in terms of dE(u, v) has received a lot of attention in the literature [1, 2, 6, 8]. In this paper, we improve these upper bounds for values of r = ω( √ logn) (i.e. 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 dG(u, v) in terms of dE(u, v).