Smith Normal Form of a distance matrix inspired by the four-point condition
Linear Algebra and Its Applications
The four point theorem is a condition for distances to arise from trees. Based on this condition, for any tree T on n vertices, we associate an (n2)×(n2) matrix MT. We find the rank and the Smith Normal Form (SNF) of the matrix MT and show that it only depends on n and is independent of the structure of the tree T. Curiously, the non-zero part of the SNF of MT coincides with the SNF of the distance matrix of T. Many such “tree independent” results are known and this result is yet another such result.
Bapat, R. B. and Sivasubramanian, Sivaramakrishnan, "Smith Normal Form of a distance matrix inspired by the four-point condition" (2020). Journal Articles. 104.