On distinct distance sets in a graph

A distinct distance set (DD set) for a graph G is a vertex subset of G with the property that for vbSvb = s, we have (2s) distinct distances of the pairs of vertices in S. In this article, it is shown that (a) For 6 ⩽ k ⩽ 18 there exists a tree T with DD(T) = k and dm(T) = LB(k) < B1(Kk), where LB(k) is the smallest value L for which there are positive integers A and B with A+B=k, A2+B2⩽⌊L2⌋, AB⩽⌈L2⌉. (b) For k ⩽ 10, the minimum order of a tree T with DD(T) = k is B1(Kk), which is the order of the corresponding Colomb ruler for k.