1. Symmetric nonnegative trifactorization of pattern matricesDamjana Kokol-Bukovšek, Helena Šmigoc, 2025, izvirni znanstveni članek Povzetek: A factorization of an $n \times n$ nonnegative symmetric matrix $A$ of the form $BCB^T$, where $C$ is a $k \times k$ symmetric matrix, and both $B$ and $C$ are required to be nonnegative, is called the Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization). The SNT-rank of $A$ is the minimal $k$ for which such factorization exists. The SNT-rank of a simple graph $G$ that allows loops is defined to be the minimal possible SNT-rank of all symmetric nonnegative matrices whose zero-nonzero pattern is prescribed by the graph $G$. We define set-join covers of graphs, and show that finding the SNT-rank of $G$ is equivalent to finding the minimal order of a set-join cover of $G$. Using this insight we develop basic properties of the SNT-rank for graphs and compute it for trees and cycles without loops. We show the equivalence between the SNT-rank for complete graphs and the Katona problem, and discuss uniqueness of patterns of matrices in the factorization.
Ključne besede: mathematics, mathematical economy, matrix algebra, nonnegative matrix factorization, nonnegative symmetric matrices, symmetric nonnegative trifactorization, pattern matrices
Objavljeno v DiRROS: 03.11.2025; Ogledov: 186; Prenosov: 98
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2. Orthogonalisability of joins of graphsRupert H. Levene, Polona Oblak, Helena Šmigoc, 2025, izvirni znanstveni članek Povzetek: A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the connected components of two graphs for their join to be orthogonalisable. In some cases, those conditions coincide, and we present several families of joins of graphs that are orthogonalisable.
Ključne besede: symmetric matrix, orthogonal matrix, inverse eigenvalue problem, minimum number of distinct eigenvalues, join of graphs
Objavljeno v DiRROS: 08.07.2025; Ogledov: 383; Prenosov: 233
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3. Bordering of symmetric matrices and an application to the minimum number of distinct eigenvalues for the join of graphsAida Abiad, Shaun M. Fallat, Mark Kempton, Rupert H. Levene, Polona Oblak, Helena Šmigoc, Michael Tait, Kevin N. Vander Meulen, 2023, izvirni znanstveni članek Ključne besede: inverse eigenvalue problem, minimum number of distinct eigenvalues, borderings, joins of graphs, paths, cycles, hypercubes
Objavljeno v DiRROS: 18.03.2024; Ogledov: 1186; Prenosov: 543
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4. The liberation set in the inverse eigenvalue problem of a graphJephian C.-H. Lin, Polona Oblak, Helena Šmigoc, 2023, izvirni znanstveni članek Povzetek: 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.
Ključne besede: symmetric matrix, inverse eigenvalue problem, strong spectral property, Matrix Liberation Lemma, zero forcing
Objavljeno v DiRROS: 14.03.2024; Ogledov: 1068; Prenosov: 569
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