Families of multiweights and pseudostars

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• 作者：Agnese Baldisserri ^{baldisser@math.unifi.it" class="auth_mail" title="E-mail the corresponding author} ; Elena Rubei ; ^{rubei@math.unifi.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C05 ; 05C12 ; 05C22
• 刊名：Advances in Applied Mathematics
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：77
• 期：Complete
• 页码：86-100
• 全文大小：397 K

文摘

Let T=(T,w)$\mathcal{T}=(T,w)$ be a weighted finite tree with leaves 1,…,n$1,\dots ,n$. For any I:={i₁,…,i_k}⊂{1,…,n}$I:=\{i_1,\dots ,i_k\}\subset \{1,\dots ,n\}$, let D_I(T)$D_I(\mathcal{T})$ be the weight of the minimal subtree of T   connecting i₁,…,i_k$i_1,\dots ,i_k$; the D_I(T)$D_I(\mathcal{T})$ are called k  -weights of T$\mathcal{T}$. Given a family of real numbers parametrized by the k  -subsets of {1,…,n}$\{1,\dots ,n\}$, baab27e2b15">${\{D_I\}}_{I\in \left(\begin{array}{c}\{1,\dots ,n\}\\ k\end{array}\right)}$, we say that a weighted tree T=(T,w)$\mathcal{T}=(T,w)$ with leaves 1,…,n$1,\dots ,n$ realizes the family if D_I(T)=D_I$D_I(\mathcal{T})=D_I$ for any I.

In [14] Pachter and Speyer proved that, if 3≤k≤(n+1)/2$3\le k\le (n+1)/2$ and baab27e2b15">${\{D_I\}}_{I\in \left(\begin{array}{c}\{1,\dots ,n\}\\ k\end{array}\right)}$ is a family of positive real numbers, then there exists at most one positive-weighted essential tree T$\mathcal{T}$ with leaves 1,…,n$1,\dots ,n$ that realizes the family (where “essential” means that there are no vertices of degree 2). We say that a tree P   is a pseudostar of kind (n,k)$(n,k)$ if the cardinality of the leaf set is n and any edge of P divides the leaf set into two sets such that at least one of them has cardinality ≥k  . Here we show that, if 3≤k≤n−1$3\le k\le n-1$ and baab27e2b15">${\{D_I\}}_{I\in \left(\begin{array}{c}\{1,\dots ,n\}\\ k\end{array}\right)}$ is a family of real numbers realized by some weighted tree, then there is exactly one weighted essential pseudostar P=(P,w)$\mathcal{P}=(P,w)$ of kind (n,k)$(n,k)$ with leaves 1,…,n$1,\dots ,n$ and without internal edges of weight 0, that realizes the family; moreover we describe how any other weighted tree realizing the family can be obtained from P$\mathcal{P}$. Finally we examine the range of the total weight of the weighted trees realizing a fixed family.