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

This article continues an investigation begun in [2]. A random graph G(n)(x) is constructed on independent random points U-1, ..., U-n distributed uniformly on [0, 1](d), d greater than or equal to 1, in which two distinct such points are joined by an edge if the l(infinity)-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 d(n), the smallest x such that G(n)(x) has no vertices of degree zero, is shown to satisfy lim(n-->infinity) (d(n)(d) n/log n) = 1/2d, d greater than or equal to 1, a.s. Series and sequence criteria on edge distances {x(n)} 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.