On the weights of simple paths in weighted complete graphs

Consider a weighted graph G with n vertices, numbered by the set {1, ..., n}. 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 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 R, there exists a weighted complete graph G such that the multisubset is equal to D_i,j (G) for some i, j vertices of G. Besides we establish a criterion to say when, given for any i, j ∈ {1, ..., n} a multisubset of R, D_i,j , there exists a weighted complete graph G with vertices {1, ..., n} such that D_i,j(G) = D_i,j for any i, j.