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For any path p in G, we call w_G(p) the sum of the weights of the edges of the path and we define the multiset {\\cal D}_{i,j} (G) = {w_G(p) | p simple path between i and j} We establish a criterion to say when, given a multisubset of the set of the real numbers there exists a weighted complete graph G such that the multisubset is equal to {\\cal D}_{i,j} (G) for some i,j vertices of G. 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