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- W4319323570 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." @default.
- W4319323570 created "2023-02-08" @default.
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- W4319323570 date "2023-02-03" @default.
- W4319323570 modified "2023-09-28" @default.
- W4319323570 title "The liberation set in the inverse eigenvalue problem of a graph" @default.
- W4319323570 doi "https://doi.org/10.48550/arxiv.2302.01670" @default.
- W4319323570 hasPublicationYear "2023" @default.
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