The maximum vertex degree of a graph on uniform points in [0,1]d

A random graph Gn(x) is constructed on independent random points U1, …, Un distributed uniformly on [0,1]^d, d ⩾ 1, in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 < x < 1. Almost-sure asymptotic results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn(x) has no vertices of degree zero, is shown to satisfy ^limn→∞(d^dnnlogn) = 1/2d, d ⩾ 1, almost surely. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.