Fori = 1,...,n letC(xi, ri) be a circle in the plane with centrexi and radiusri. A repeated distance graph is a directed graph whose vertices are the centres and where (xi, xj) is a directed edge wheneverxj lies on the circle with centrexi. Special cases are the nearest neighbour graph, whenri is the minimum distance betweenxi 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 ind dimensions, with particularly tight bounds for the furthest neighbour graph in three dimensions. The proofs use extremal graph theory.
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