Weighted Graphs with Distances in Given Ranges

Abstract

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.