2014/06/30 by Baldisserri, Agnese
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1407.0048
Let \cal T=(T,w) be a positive-weighted tree with at least n vertices. For any i,j ∈ \1,...,n\, let Di,j (\cal T) be the weight of the unique path in T connecting i and j. The Di,j (\cal T) are called 2-weights of \cal T and, if we put in order the 2-weights, the vector which has the Di,j (\cal T) as components is called 2-dissimilarity vector of \cal T. Given a family of positive real numbers \Di,j\_i,j ∈ \1,...,n\, we say that a positive-weighted tree \cal T=(T,w) realizes the family if \1,...,n\ ⊂ V(T) and Di,j(\cal T)=Di,j for any i,j ∈ \1,...,n\. A characterization of 2-dissimilarity families of positive weighted trees is already known (see \citeB, \citeSimP or \citeSt): the families must satisfy the well-known four-point condition. However we can wonder when there exists a positive-weighted tree with exactly n vertices, 1,...,n, and realizing the family \Di,j\. In this paper we will show that the four-point condition is necessary but no more sufficient, and so we will introduce two additional conditions (see Theorem \refthm:ThmAgne).