Repeated distances in space

For i = 1, . . . . n let C(x;, r,) be a circle in the plane with centre .x i and radius r; . A repeated distance graph is a directed graph whose vertices are the centres and where (xi, x;) is a directed edge whenever x; lies on the circle with centre x, . Special cases are the nearest neighbour graph, when ri is the minimum distance between x, and any other centre, and the furthest neighbour graph which is similar except that maximum replaces minimum . Repeated distance graphs generalize to any dimension with spheres or hyperspheres replacing circles. Bounds are given on the number of edges in repeated distance graphs in d dimensions, with particularly tight bounds for the furthest neighbour graph in three dimensions . The proofs use extremal graph theory .