The maximum four point condition matrix of a tree

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 $\Max_T$.