$\newcommand{\Max}{\mathrm{Max4PC}}$ 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 $\Max_T$.

翻译：$\newcommand{\Max}{\mathrm{Max4PC}}$ 四点条件（以下简称4PC）是刻画树$T$中距离的一个众所周知的条件。设$w,x,y,z$为树$T$中的四个顶点，$d_{x,y}$表示顶点$x$与$y$在$T$中的距离。4PC条件指出，在$d_{w,x} + d_{y,z}$、$d_{w,y} + d_{x,z}$和$d_{w,z} + d_{x,y}$这三个项中，最大值等于次大值。我们定义了一个大小为$\binom{n}{2} \times \binom{n}{2}$的矩阵$\Max_T$，其行和列均由大小为2的子集索引。$\Max_T$中对应于行索引$\{w,x\}$、列索引$\{y,z\}$的条目，是$d_{w,x} + d_{y,z}$、$d_{w,y} + d_{x,z}$和$d_{w,z} + d_{x,y}$这三个项中的最大值。在本文中，我们确定了该矩阵的基本性质（如秩），给出了一种输出基族的算法，并求出了$\Max_T$限制于我们基下的行列式。此外，我们进一步确定了$\Max_T$的惯性指数和史密斯标准型（SNF）。