多重权的类图族
Graphlike families of multiweights
摘要：
设${\cal G}=(G,w)$是一个赋权图，即赋予从$G$的边集到实数集的函数$w$的图$G$;对于$G$的顶点集的任何子集$S$，我们定义$D_S({\cal G})$为$G$的顶点集包含$S$的子图的权的最小值;我们称$D_S({\cal G})$为${\cal G}$的多重权。设$X$是有限集，$\{D_S\}_{S \subset X, \; \sharp S \geq 2} $是正实数族.我们找到了这个族是顶点集为$X$的正赋权图的重权族的充分必要条件。此外，我们还研究了树的类似问题。最后，我们找到了一个标准，如果存在一个非负加权树${\cal T}$与叶集$X$，使$D_S ({\cal T})=D_S $的任何$S \subset X$。
英文摘要：
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 $S \subset X$.
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