Let {cal T}=(T,w) be a weighted finite tree with leaves 1,..., n. For any I :={i_1,..., i_k } subset of {1,...,n}, let D_I ({cal T}) be the weight of the minimal subtree of T connecting i_1,..., i_k; the D_{I} ({cal T}) are called k-weights of {cal T}. Given a family of real numbers parametrized by the k-subsets of {1,..., n}, {D_I}_{I in {{1,...,n} choose k}}, we say that a weighted tree {cal T}=(T,w) with leaves 1,..., n realizes the family if D_I({cal T})=D_I for any I. In cite{P-S} Pachter and Speyer proved that, if 3 <= k l<= (n+1)/2 and {D_I}_{I in {{1,...,n} choose k}} is a family of positive real numbers, then there exists at most one positive-weighted essential tree {cal T} with leaves 1,...,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) 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 and {D_I}_{I in {{1,...,n} choose k}} is a family of real numbers realized by some weighted tree, then there is exactly one weighted essential pseudostar {cal P}=(P,w) of kind (n,k) with leaves 1,...,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 {cal P}. Finally we examine the range of the total weight of the weighted trees realizing a fixed family.
Families of multiweights and pseudostars / Baldisserri, Agnese; Rubei, Elena. - In: ADVANCES IN APPLIED MATHEMATICS. - ISSN 0196-8858. - STAMPA. - 77:(2016), pp. 86-100. [10.1016/j.aam.2016.03.001]
Families of multiweights and pseudostars
BALDISSERRI, AGNESE;RUBEI, ELENA
2016
Abstract
Let {cal T}=(T,w) be a weighted finite tree with leaves 1,..., n. For any I :={i_1,..., i_k } subset of {1,...,n}, let D_I ({cal T}) be the weight of the minimal subtree of T connecting i_1,..., i_k; the D_{I} ({cal T}) are called k-weights of {cal T}. Given a family of real numbers parametrized by the k-subsets of {1,..., n}, {D_I}_{I in {{1,...,n} choose k}}, we say that a weighted tree {cal T}=(T,w) with leaves 1,..., n realizes the family if D_I({cal T})=D_I for any I. In cite{P-S} Pachter and Speyer proved that, if 3 <= k l<= (n+1)/2 and {D_I}_{I in {{1,...,n} choose k}} is a family of positive real numbers, then there exists at most one positive-weighted essential tree {cal T} with leaves 1,...,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) 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 and {D_I}_{I in {{1,...,n} choose k}} is a family of real numbers realized by some weighted tree, then there is exactly one weighted essential pseudostar {cal P}=(P,w) of kind (n,k) with leaves 1,...,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 {cal P}. Finally we examine the range of the total weight of the weighted trees realizing a fixed family.File | Dimensione | Formato | |
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pseudostars.pdf
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