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Let G$$ \\mathcal{G} $$ = (G,w) be a weighted simple finite connected graph, that is, let G be a simple finite connected graph endowed with a function w from the set of the edges of G to the set of real numbers. For any subgraph G¿ of G, we define w(G¿) to be the sum of the weights of the edges of G¿. For any i, j vertices of G, we define D{i,j}(G$$ \\mathcal{G} $$) to be the minimum of the weights of the simple paths of G joining i and j. The D{i,j}(G$$ \\mathcal{G} $$) are called 2-weights of G$$ \\mathcal{G} $$. Weighted graphs and their reconstruction from 2-weights have applications in several disciplines, such as biology and psychology.Let mII¿1¿n2$$ {\\left\\{{m}_I\\right\\}}_{I\\in \\left(\\frac{\\left\\{1,\\dots, n\\right\\}}{2}\\right)} $$ and MII¿1¿n2$$ {\\left\\{{M}_I\\right\\}}_{I\\in \\left(\\frac{\\left\\{1,\\dots, n\\right\\}}{2}\\right)} $$ be two families of positive real numbers parametrized by the 2-subsets of {1, ¿, n} with mI ≤ MI for any I; we study when there exist a positive-weighted graph G and an n-subset {1, ¿, n} of the set of its vertices such that DI (G$$ \\mathcal{G} $$) ¿ [mI,MI] for any I¿1¿n2$$ I\\in \\left(\\frac{\\left\\{1,\\dots, n\\right\\}}{2}\\right) $$. Then we study the analogous problem for trees, both in the case of positive weights and in the case of general weights.

Weighted Graphs with Distances in Given Ranges