Enhanced negative type for finite metric trees

摘要

A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of the non-trivial 1-negative type inequalities for finite metric trees. These inequalities are sufficiently strong to imply that any given finite metric tree (T, d) must have strict p-negative type for all p in an open interval (1-zeta, 1+zeta), where zeta > 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on T. In particular, if the edges of the tree are not weighted, then it follows that depends only upon the number of vertices in the tree. We also give an example of an infinite metric tree that has strict 1-negative type but does not have p-negative type for any p > 1. This shows that the maximal p-negative type of a metric space can be strict. (C) 2008 Published by Elsevier Inc.