Graphlike families of multiweights

TL;DR

This paper characterizes when a family of positive real numbers can be realized as multiweights of weighted graphs and trees, providing necessary and sufficient conditions for such representations.

Contribution

It establishes criteria for families of numbers to correspond to multiweights of weighted graphs and trees, including nonnegative-weighted trees with specified leaf sets.

Findings

Characterization of multiweights for weighted graphs

Criteria for multiweights in trees

Existence conditions for nonnegative-weighted trees

Abstract

Let ${\cal G}=(G,w)$ be a weighted graph , that is, a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of real numbers; for any subset $S$ of the vertex set of $G$, we define $D_S({\cal G})$ to be the minimum of the weights of the subgraphs of $G$ whose vertex set contains $S$; we call $D_S({\cal G})$ a multiweight of ${\cal G}$. Let $X$ be a finite set and let $\{D_S\}_{S \subset X, \; \sharp S \geq 2} $ be a family of positive real numbers. We find necessary and sufficient conditions for the family to be the family of multiweights of a positive-weighted graph with vertex set $X$. Moreover we study the analogous problem for trees. Finally, we find a criterion to say if there exists a nonnegative-weighted tree ${\cal T}$ with leaf set $X$ and such that $D_S ({\cal T})=D_S $ for any $S \subset X$.