Weighted graphs with distances in given ranges

Let $\mathcal{G}=(G,w)$ 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 $G^{\mathrm{\prime }}$ of $G$, we define $w(G^{\mathrm{\prime }})$ to be the sum of the weights of the edges of $G^{\mathrm{\prime }}$. For any $i,j$ vertices of $G$, we define $D_{\{i,j\}}(\mathcal{G})$ to be the minimum of the weights of the simple paths of $G$ joining $i$ and $j$. The $D_{\{i,j\}}(\mathcal{G})$ are called $2$-weights of $\mathcal{G}$. Let $\{m_I{\}}_{I\in \left(\genfrac{}{}{0px}{}{\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}}{2}\right)}$ and ${M_I}_{I \in {{1,…,n} \choose 2}}$ be two families of positive real numbers parametrized by the $2$-subsets of $\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ with $m_I\le M_I$ for any $I$; we study when there exist a positive-weighted graph $\mathcal{G}$ and an $n$-subset $\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ of the set of its vertices such that $D_I ({\cal G}) \in [m_I, M_I] $forany$I \in {{1,…,n} \choose 2}$. Then we study the analogous problem for trees, both in the case of positive weights and in the case of general weights.