Symmetric Matrices Representable by Weighted Trees over a Cancellative Abelian Monoid
Hans‐Jürgen Bandelt, Michael Anthony Steel · SIAM Journal on Discrete Mathematics · 1995
The classical result that characterizes metrics induced by paths in a labeled tree having positive real edge weights is generalized to allow the edge weights to take values in any cancellative abelian monoid satisfying the additional requirement that $x + x = y + y$ implies $x = y$. This includes the case of arbitrary real-valued edge weights, which applies to distance-hereditary graphs, thus yielding (unique) weighted tree representations for the latter.