Four-point condition matrices of edge-weighted trees

Abstract

Formulas for the determinant of distance matrix DT{D}_{T} of tree TT are known in the unweighted case and in the case when the edges of TT have commuting variable weights. Associated with the four-point condition (4PC) and a tree TT are two matrices, the Max4PCT{{\rm{Max4PC}}}_{T} and the Min4PCT{{\rm{Min4PC}}}_{T}. These are not full rank matrices and their rank, a basis BB, and formulas for the determinant when restricted to the rows and columns of BB are known. In this work, we generalize both these matrices to the case when the edges of TT have commuting variable weights and determine edge-weighted counterparts of known results.

Keywords

• distance matrix
• tree
• four-point condition
• edge weights
• 05c05
• 05c12
• secondary 15a15