The liberation set in the inverse eigenvalue problem of a graph

Lin, Jephian C.-H. (avtor), Oblak, Polona (avtor), Šmigoc, Helena (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: symmetric matrix, inverse eigenvalue problem, strong spectral property, Matrix Liberation Lemma, zero forcing
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FRI - Fakulteta za računalništvo in informatiko; FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2023
• Št. strani: Str. 1-28
• Številčenje: Vol. 675
• PID: 20.500.12556/RUL-147499
• UDK: 512
• ISSN pri članku: 0024-3795
• DOI: 10.1016/j.laa.2023.06.009
• COBISS.SI-ID: 157762051
• Datum objave v RUL: 06.07.2023

LIN, Jephian C.-H., OBLAK, Polona in ŠMIGOC, Helena, 2023, The liberation set in the inverse eigenvalue problem of a graph. Linear algebra and its applications [na spletu]. 2023. Vol. 675, p. 1–28. [Dostopano 6 april 2025]. DOI 10.1016/j.laa.2023.06.009. Pridobljeno s: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=slv&id=147499

Gradivo je del revije

• Naslov: Linear algebra and its applications
• Skrajšan naslov: Linear algebra appl.
• Založnik: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

Licence

• Licenca: CC BY-NC-ND 4.0, Creative Commons Priznanje avtorstva-Nekomercialno-Brez predelav 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by-nc-nd/4.0/deed.sl
• Opis: Najbolj omejujoča licenca Creative Commons. Uporabniki lahko prenesejo in delijo delo v nekomercialne namene in ga ne smejo uporabiti za nobene druge namene.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: simetrična matrika, inverzni problem lastnih vrednosti, krepka spektralna lastnost, ničelna prisila