20.500.12556/DiRROS-18403 The liberation set in the inverse eigenvalue problem of a graph 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. symmetric matrix inverse eigenvalue problem strong spectral property Matrix Liberation Lemma zero forcing simetrična matrika inverzni problem lastnih vrednosti krepka spektralna lastnost ničelna prisila true false true Angleški jezik Slovenski jezik Neznano 2024-03-14 13:45:44 2024-03-14 13:45:44 2024-03-15 03:36:23 0000-00-00 00:00:00 2023 0 0 str. 1-28 Vol. 675 Oct. 2023 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2023-10-01 512 0024-3795 10.1016/j.laa.2023.06.009 157762051 https://www.sciencedirect.com/science/article/pii/S0024379523002276?via=ihub 1 c69e9d99-e200-11ee-9dd4-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=25059 1-s2.0-S0024379523002276-main.pdf 1-s2.0-S0024379523002276-main.pdf 1 16393A6DD827E34C97CD6D0EF83BBE11 4bdedb489d931613ef6bd46dd923d7f097125f9dd132d4d7bc537ad02f75b632 f0be1d23-e200-11ee-9dd4-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=25060 Inštitut za matematiko, fiziko in mehaniko 0 0 0