Scaffoldings of Totally Positive Matrices and Line Insertion

TL;DR

This paper explores how to insert a line into a totally positive matrix while preserving total positivity, using graph-based representations to characterize all possible insertions.

Contribution

It introduces a graph-theoretic framework called scaffolding to characterize all line insertions maintaining total positivity.

Findings

Every totally positive matrix can be associated with a vertex-weighted graph.

All possible line insertions correspond to strongly positive solutions of a linear system.

Provides a complete characterization of line insertions in totally positive matrices.

Abstract

Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work we revisit this problem. First we show that every totally positive matrix can be associated to a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.