arXiv 1409.38632014-09math.CO

距离在给定范围内的加权图

Weighted graphs with distances in given ranges

Elena Rubei

备注：: Comments are welcome; some minor changes (thm.16)

摘要：

设${\cal G}=(G,w)$是一个加权简单有限连通图，也就是说，设$G$是一个简单有限连通图，它具有一个从$G$的边集到实数集的函数$w$.对于$G$的任何子图$G'$，我们定义$w(G')$为$G'$的边的权重之和。对于$G$的任何$i,j $顶点，我们定义$D_{\{i,j\}} ({\cal G})$为连接$i$和$j$的$G$的简单路径的权重的最小值。$D_{\{i,j\}} ({\cal G})$称为$2$-${\cal G}$的权重。设$\{m_I\}_{I \in {\{1,...,n\} \choose 2}}$和$\{M_I\}_{I \in {\{1,...,n\} \choose 2}}$是由$ \{1,..., n\}$的$2$ -子集参数化的两个正实数族，对任何$I$都有$m_I \leq M_I$;我们研究了当存在正权图${\cal G}$和它的顶点集的$n$ -子集$\{1,..., n\}$，使得对任何$I \in {\{1,...,n\} \choose 2}$都有$D_I ({\cal G}) \in [m_I, M_I] $.然后，我们研究了类似的问题，树的情况下，在正的重量和一般的重量。

英文摘要：

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

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摘要：

英文摘要：

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