The maximum four point condition matrix of a tree

08/16/2023

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by   Ali Azimi, et al.

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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 n2×n2 sized matrix _T from a tree T where the rows and columns are indexed by size-2 subsets. The entry of _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 _T when restricted to our basis. We further determine the inertia and the Smith Normal Form (SNF) of _T.