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.
2016
77
86
100
Goal 17: Partnerships for the goals
Baldisserri, Agnese; Rubei, Elena
File in questo prodotto:
File Dimensione Formato  
pseudostars.pdf

Open Access dal 02/07/2018

Descrizione: .
Tipologia: Versione finale referata (Postprint, Accepted manuscript)
Licenza: Tutti i diritti riservati
Dimensione 298.88 kB
Formato Adobe PDF
298.88 kB Adobe PDF
24-B-R-AAM-families.pdf

Accesso chiuso

Descrizione: Articolo principale
Tipologia: Pdf editoriale (Version of record)
Licenza: Tutti i diritti riservati
Dimensione 397.6 kB
Formato Adobe PDF
397.6 kB Adobe PDF   Richiedi una copia

I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1029150
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? 6
social impact