SummaryRandom associations of points on a graphLet be given a graph consisting of n points, numbered from 1, …, n, and mijjoins between points i and j. From this graph r points are chosen at random. This paper deals with the properties of a statisticx, being the number of joins between chosen points. Applications are considered to a test for randomness, and to the order‐disorder problem.Analogous results for other statistics and also for the case in which the points are sampled independently of each other, are announced. These results are described in Bloemena, (19