The liberation set in the inverse eigenvalue problem of a graph

TL;DR

This paper refines the Matrix Liberation Lemma for the inverse eigenvalue problem of a graph, enabling better analysis of eigenvalues in matrices related to graph structures, and applies it to solve open cases for six-vertex graphs.

Contribution

We provide an equivalent form of the Matrix Liberation Lemma, simplifying its application and extending its use to direct sums of graphs, revealing a connection with zero forcing games.

Findings

The new lemma reduces technical difficulties in applying the Matrix Liberation Lemma.

Application to matrices of the form M=A⊕B demonstrates enhanced results beyond the strong spectral property.

Resolved several open cases in the inverse eigenvalue problem for graphs with six vertices.

Abstract

The inverse eigenvalue problem of a graph $G$ is the problem of characterizing all lists of eigenvalues of real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of $G$. The strong spectral property is a powerful tool in this problem, which identifies matrices whose entries can be perturbed while controlling the pattern and preserving the eigenvalues. The Matrix Liberation Lemma introduced by Barrett et al.~in 2020 advances the notion to a more general setting. In this paper we revisit the Matrix Liberation Lemma and prove an equivalent statement, that reduces some of the technical difficulties in applying the result. We test our method on matrices of the form $M=A \oplus B$ and show how this new approach supplements the results that can be obtained from the strong spectral property only. While extending this notion to the direct sums of graphs, we discover a surprising connection with the zero forcing game on Cartesian products of graphs. Throughout the paper we apply our results to resolve a selection of open cases for the inverse eigenvalue problem of a graph on six vertices.