On the relation between weighted trees and tropical Grassmannians

TL;DR

This paper establishes a connection between 4-dissimilarity vectors of n-trees and tropical Grassmannians, proposing conjectures for higher dimensions and confirming some cases computationally.

Contribution

It proves the inclusion of 4-dissimilarity vectors in tropical Grassmannians and introduces conjectures for higher dimensions, verified for m=5 using computer algebra.

Findings

4-dissimilarity vectors are contained in G_{4,n}

Three equivalent conjectures for m > 4

Conjectures verified for m=5 with computer algebra

Abstract

In this article, we will prove that the set of 4-dissimilarity vectors of n-trees is contained in the tropical Grassmannian G_{4,n}. We will also propose three equivalent conjectures related to the set of m-dissimilarity vectors of n-trees for the case m > 4. Using a computer algebra system, we can prove these conjectures for m = 5.