The maximum four point condition matrix of a tree

TL;DR

This paper introduces a matrix derived from a tree's distances, explores its algebraic properties such as rank, determinant, inertia, and Smith Normal Form, and provides algorithms related to its bases.

Contribution

It defines the maximum four point condition matrix for trees and analyzes its algebraic properties, including rank, bases, determinant, inertia, and SNF.

Findings

Determined the rank of the matrix for trees.

Provided an algorithm for bases of the matrix.

Calculated the determinant, inertia, and Smith Normal Form.

Abstract

$\newcommand{\Max}{\mathrm{Max4PC}}$ The Four point condition (4PC henceforth) is a well known condition characterising distances in trees $T$. Let $w,x,y,z$ be four vertices in $T$ and let $d_{x,y}$ denote the distance between vertices $x,y$ in $T$. The 4PC condition says that among the three terms $d_{w,x} + d_{y,z}$, $d_{w,y} + d_{x,z}$ and $d_{w,z} + d_{x,y}$ the maximum value equals the second maximum value. We define an $\binom{n}{2} \times \binom{n}{2}$ sized matrix $\Max_T$ from a tree $T$ where the rows and columns are indexed by size-2 subsets. The entry of $\Max_T$ corresponding to the row indexed by $\{w,x\}$ and column $\{y,z\}$ is the maximum value among the three terms $d_{w,x} + d_{y,z}$, $d_{w,y} + d_{x,z}$ and $d_{w,z} + d_{x,y}$. In this work, we determine basic properties of this matrix like rank, give an algorithm that outputs a family of bases, and find the determinant of $\Max_T$ when restricted to our basis. We further determine the inertia and the Smith Normal Form (SNF) of $\Max_T$.