SYMMETRICAL MATRICES REPRESENTABLE BY WEIGHTED TREES OVER A CANCELLATIVE ABELIAN MONOID

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.