  Given any two vertices u, v of a random geometric graph, denote by d_E(u,v)
their Euclidean distance and by d_G(u,v) their graph distance. The problem of
finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of
attention in the literature. In this paper, we improve these upper bounds for
values of r=omega(sqrt(log n)) (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 d_G(u,v) in terms of
d_E(u,v).
