A tropical interpretation of m-dissimilarity maps

TL;DR

This paper introduces a tropical geometric approach to compute and describe m-dissimilarity maps of weighted trees, providing formulas and characterizations for specific cases like m=3 and partial results for m=4.

Contribution

It presents a novel tropical formula for calculating m-dissimilarity maps from distance matrices of trees, advancing the mathematical understanding of tree metrics.

Findings

Tropical formula for m-dissimilarity maps for any m between 2 and n.

Tropical description of the set of 3-dissimilarity maps.

Partial results for 4-dissimilarity maps.

Abstract

Let T be a weighted tree with n numbered leaves and let D be its distance matrix, so D(i,j) is the distance between the leaves i and j. If m is an integer between 2 and n, we prove a tropical formula to compute the m-dissimilarity map of T (i.e. the weights of the subtrees of T with m leaves), given D. For m equal to 3, we present a tropical description of the set of m-dissimilarity maps of trees. For m equal to 4, a partial result is given.