On the weights of simple paths in weighted complete graphs

Abstract

Consider a weighted simple graph G on the vertex set {1,..., n}. For any path p in G, we call ω _G (p) the sum of the weights of the edges of the path, and, for any {i, j} ⊂ {1,..., n}, we define the multiset

$$D_{\{ i,j\} } (G) = \left\{ {\left. {w_G \left( p \right)} \right|p a simple path between i and j} \right\}.$$

We establish a criterion for a multisubset of ℝ to be of the form $\mathcal{D}_{\{ i,j\} } (G)$ for some weighted complete graph G and for some i, j vertices of G.

Besides we establish a criterion for a family

of multi-subsets of ℝ to be of the form

for some weighted complete graph G with vertex set {1,..., n}.