{"title":"Combinatorial explanation of coefficients of the signless Laplacian characteristic polynomial of a digraph","authors":"Jingyuan Zhang , Xian'an Jin , Weigen Yan","doi":"10.1016/j.laa.2025.03.010","DOIUrl":"10.1016/j.laa.2025.03.010","url":null,"abstract":"<div><div>Let <em>G</em> be a simple digraph with <em>n</em> vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Denote the adjacency matrix and the in-degree matrix of <em>G</em> by <span><math><mi>A</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span> and <span><math><mi>D</mi><mo>=</mo><mi>d</mi><mi>i</mi><mi>a</mi><mi>g</mi><mo>(</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mo>,</mo><mo>⋯</mo><mo>,</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></math></span>, respectively, where <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></math></span> if <span><math><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span> is an arc of <em>G</em> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span> otherwise, and <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> is the number of arcs with head <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in <em>G</em>. Set <span><math><mi>f</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>det</mi><mo></mo><mo>(</mo><mi>x</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>D</mi><mo>−</mo><mi>A</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>i</mi></mrow></msup><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msup></math></span>, where <span><math><mi>det</mi><mo></mo><mo>(</mo><mi>X</mi><mo>)</mo></math></span> denotes the determinant of a square matrix <em>X</em>. Then <span><math><mi>f</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span> is called the signless Laplacian characteristic polynomial of the digraph <em>G</em>. Li, Lu, Wang and Wang (2023) <span><span>[7]</span></span> gave a combinatorial explanation of <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> of <span><math><mi>f</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>x</mi><mo>)</mo></math></span>. In this paper, we give combinatorial explanations of all the coefficient","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"717 ","pages":"Pages 56-67"},"PeriodicalIF":1.0,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the limit points of the smallest positive eigenvalues of graphs","authors":"Sasmita Barik, Debabrota Mondal","doi":"10.1016/j.laa.2025.03.006","DOIUrl":"10.1016/j.laa.2025.03.006","url":null,"abstract":"<div><div>In 1972, Hoffman <span><span>[11]</span></span> initiated the study of limit points of eigenvalues of nonnegative symmetric integer matrices. He posed the question of finding all limit points of the set of spectral radii of all nonnegative symmetric integer matrices. In the same article, the author demonstrated that it is enough to consider the adjacency matrices of simple graphs to study the limit points of spectral radii. Since then, many researchers have worked on similar problems, considering various specific eigenvalues such as the least eigenvalue, the <em>k</em>th largest eigenvalue, and the <em>k</em>th smallest eigenvalue, among others. Motivated by this, we ask the question, “which real numbers are the limit points of the set of the smallest positive eigenvalues (respectively, the largest negative eigenvalues) of graphs?” In this article, we provide a complete answer to this question by proving that any nonnegative (respectively, nonpositive) real number is a limit point of the set of all smallest positive eigenvalues (respectively, largest negative eigenvalues) of graphs. We also show that the union of the sets of limit points of the smallest positive eigenvalues and the largest negative eigenvalues of graphs is dense in <span><math><mi>R</mi></math></span>, the set of all real numbers.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"715 ","pages":"Pages 1-16"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unitary similarity and the numerical radius preservers","authors":"Abdellatif Bourhim , Mohamed Mabrouk","doi":"10.1016/j.laa.2025.03.005","DOIUrl":"10.1016/j.laa.2025.03.005","url":null,"abstract":"<div><div>Let <span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> be the algebra of all bounded linear operators acting on a separable infinite-dimensional complex Hilbert space <span><math><mi>H</mi></math></span>, and denote by <span><math><mi>w</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> the numerical radius of any operator <span><math><mi>T</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>. In this paper, we describe the form of all bijective linear maps <em>ϕ</em> on <span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> for which <span><math><mi>w</mi><mo>(</mo><mi>ϕ</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>w</mi><mo>(</mo><mi>ϕ</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>)</mo></math></span> whenever <span><math><mi>T</mi><mo>,</mo><mspace></mspace><mi>S</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> are two unitarily similar operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 15-27"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Christensen-Sinclair factorization via semidefinite programming","authors":"Francisco Escudero-Gutiérrez","doi":"10.1016/j.laa.2025.03.007","DOIUrl":"10.1016/j.laa.2025.03.007","url":null,"abstract":"<div><div>We show that the Christensen-Sinclair factorization theorem, when the underlying Hilbert spaces are finite dimensional, is an instance of strong duality of semidefinite programming. This gives an elementary proof of the result and also provides an efficient algorithm to compute the Christensen-Sinclair factorization.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 28-44"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interval global optimization problem in max-plus algebra","authors":"Helena Myšková, Ján Plavka","doi":"10.1016/j.laa.2025.03.009","DOIUrl":"10.1016/j.laa.2025.03.009","url":null,"abstract":"<div><div>Consider the global optimization problem of minimizing the max-plus product <span><math><mi>A</mi><mo>⊗</mo><mi>x</mi></math></span>, where <em>A</em> is a given matrix and the constraint set is the set of column vectors <em>x</em> such that the sum of products <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub><mspace></mspace><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> is equal to <em>c</em> and <em>c</em> is a given positive real constant, <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> are non-negative numbers with sum equal to 1. We show that the solvability of the given global optimization problem is independent of the number <em>c</em> if the components of the vector <em>x</em> can also be negative. From a practical point of view, we further consider the solvability of the global optimization problem with non-negative constraints. We propose an algorithm which decides whether a given problem is solvable, extend the problem to interval matrices and provide an algorithm to verify the solvability of interval global optimization problem.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 45-63"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Isometries of the qubit state space with respect to quantum Wasserstein distances","authors":"Richárd Simon , Dániel Virosztek","doi":"10.1016/j.laa.2025.03.004","DOIUrl":"10.1016/j.laa.2025.03.004","url":null,"abstract":"<div><div>In this paper we study isometries of quantum Wasserstein distances and divergences on the quantum bit state space. We describe isometries with respect to the symmetric quantum Wasserstein divergence <span><math><msub><mrow><mi>d</mi></mrow><mrow><mtext>sym</mtext></mrow></msub></math></span>, the divergence induced by all of the Pauli matrices. We also give a complete characterization of isometries with respect to <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>z</mi></mrow></msub></math></span>, the quantum Wasserstein distance corresponding to the single Pauli matrix <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>z</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"714 ","pages":"Pages 1-14"},"PeriodicalIF":1.0,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Structured backward errors for special classes of saddle point problems with applications","authors":"Sk. Safique Ahmad, Pinki Khatun","doi":"10.1016/j.laa.2025.03.003","DOIUrl":"10.1016/j.laa.2025.03.003","url":null,"abstract":"<div><div>In the realm of numerical analysis, the study of structured backward errors (<em>BEs</em>) in saddle point problems (<em>SPPs</em>) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the coefficient matrix of the <em>SPP</em>. Moreover, the existing techniques are not applicable when the block matrices have <em>circulant</em>, <em>Toeplitz</em>, or <em>symmetric</em>-<em>Toeplitz</em> structures and do not even provide structure-preserving minimal perturbation matrices for which the <em>BE</em> is attained. To overcome these limitations, we investigate the structured <em>BEs</em> of <em>SPPs</em> when the perturbation matrices exploit the sparsity pattern as well as <em>circulant</em>, <em>Toeplitz</em>, and <em>symmetric</em>-<em>Toeplitz</em> structures. Furthermore, we construct minimal perturbation matrices that preserve the sparsity pattern and the aforementioned structures. Applications of the developed frameworks are utilized to compute <em>BEs</em> for the weighted regularized least squares problem. Finally, numerical experiments are performed to validate our findings, showcasing the utility of the obtained structured <em>BEs</em> in assessing the strong backward stability of numerical algorithms.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"713 ","pages":"Pages 90-112"},"PeriodicalIF":1.0,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143631968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The full P-vertex problem for unicyclic graphs","authors":"A. Howlader, P.R. Raickwade, K.C. Sivakumar","doi":"10.1016/j.laa.2025.02.024","DOIUrl":"10.1016/j.laa.2025.02.024","url":null,"abstract":"<div><div>Let <em>A</em> be a real symmetric and nonsingular matrix and <em>G</em> be the underlying graph. Let <span><math><mi>A</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></span> be the principal submatrix obtained by removing the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mtext>th</mtext></mrow></msup></math></span> row and the <span><math><msup><mrow><mi>i</mi></mrow><mrow><mtext>th</mtext></mrow></msup></math></span> column of <em>A</em>. If the nullity of <span><math><mi>A</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></span> is unity, then the vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is called a P-vertex of the matrix <em>A</em>. The full P-vertex problem is to determine if there is a nonsingular matrix <em>A</em> such that each vertex of the corresponding graph <em>G</em>, is a P-vertex of <em>A</em>. In this article, we investigate the full P-vertex problem for unicyclic graphs.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"713 ","pages":"Pages 74-89"},"PeriodicalIF":1.0,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143631967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Interplay between discretization and controllability of linear delay systems: An algebraic viewpoint","authors":"Florentina Nicolau , Hugues Mounier , Silviu-Iulian Niculescu","doi":"10.1016/j.laa.2025.02.025","DOIUrl":"10.1016/j.laa.2025.02.025","url":null,"abstract":"<div><div>In this paper, we give an in depth study of linear delay systems controllability preservation/alteration through discretization. We make use of a module theoretic framework acting as a unifying one for most of the existing delay system controllability notions. We propose a formal generic definition of a discretization scheme and illustrate through examples that controllability properties may be lost through discretization. Then, we introduce the notion of preservation (that is, a measure of quantifying the ability of the discretizer to preserve controllability properties) and prove that for a given discretizer, we can always find a delay system for which even the torsion-free controllability (which is the weakest controllability notion) is not preserved. Finally, we reverse the situation, and show that for any given delay system, preserving discretizers exist.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"713 ","pages":"Pages 18-73"},"PeriodicalIF":1.0,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-generation of traceless matrices over finite fields","authors":"Omer Cantor , Urban Jezernik , Andoni Zozaya","doi":"10.1016/j.laa.2025.02.023","DOIUrl":"10.1016/j.laa.2025.02.023","url":null,"abstract":"<div><div>We prove that the Lie algebra <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> of traceless matrices over a finite field of characteristic <em>p</em> can be generated by 2 elements with exceptions when <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is <span><math><mo>(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> or <span><math><mo>(</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. In the latter cases, we establish curious identities that obstruct 2-generation.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"713 ","pages":"Pages 1-17"},"PeriodicalIF":1.0,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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