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).

Consider two independent Goldstein-Kac telegraph processes X1(t) and X2(t) on the real line ℝ. The processes Xk(t), k = 1, 2, describe stochastic motions at finite constant velocities c1 > 0 and c2 > 0 that start at the initial time instant t = 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1 > 0 and λ2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X1(t) - X2(t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.