Matrices Totally Positive Relative to a Tree, II

TL;DR

This paper proves that for a general tree, T-TP matrices with positive determinant and certain submatrix properties have smallest eigenvalues with eigenvectors signed according to the tree structure.

Contribution

It establishes a new spectral property of T-TP matrices related to eigenvector signs in the context of general trees.

Findings

Smallest eigenvalue eigenvector signs follow the tree structure.

Submatrices associated with pendant vertices are P-matrices.

Positive determinant ensures eigenvector sign pattern.

Abstract

In this paper we prove that for a general tree $T$, if $A$ is T-TP, all the submatrices of $A$ associated with the deletion of pendant vertices are $P$-matrices, and $\det A>0$, then the smallest eigenvalue has an eigenvector signed according to $T$.