{"title":"Smith forms of matrices in Companion Rings, with group theoretic and topological applications","authors":"Vanni Noferini , Gerald Williams","doi":"10.1016/j.laa.2024.12.003","DOIUrl":"10.1016/j.laa.2024.12.003","url":null,"abstract":"<div><div>Let <em>R</em> be a commutative ring and <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>[</mo><mi>t</mi><mo>]</mo></math></span> a monic polynomial. The commutative ring of polynomials <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>)</mo></math></span> in the companion matrix <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> of <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, where <span><math><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>[</mo><mi>t</mi><mo>]</mo></math></span>, is called the Companion Ring of <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>. Special instances include the rings of circulant matrices, skew-circulant matrices, pseudo-circulant matrices, or lower triangular Toeplitz matrices. When <em>R</em> is an Elementary Divisor Domain, we develop new tools for computing the Smith forms of matrices in Companion Rings. In particular, we obtain a formula for the second last non-zero determinantal divisor, we provide an <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>)</mo><mo>↔</mo><mi>g</mi><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>)</mo></math></span> swap theorem, and a composition theorem. When <em>R</em> is a principal ideal domain we also obtain a formula for the number of non-unit invariant factors. By applying these to families of circulant matrices that arise as relation matrices of cyclically presented groups, in many cases we compute the groups' abelianizations. When the group is the fundamental group of a three dimensional manifold, this provides the homology of the manifold. In other cases we obtain lower bounds for the rank of the abelianization and record consequences for finiteness or solvability of the group, or for the Heegaard genus of a corresponding manifold.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 372-404"},"PeriodicalIF":1.0,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165098","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":"The truncated univariate rational moment problem","authors":"Rajkamal Nailwal , Aljaž Zalar","doi":"10.1016/j.laa.2024.12.009","DOIUrl":"10.1016/j.laa.2024.12.009","url":null,"abstract":"<div><div>Given a closed subset <em>K</em> in <span><math><mi>R</mi></math></span>, the rational <em>K</em>–truncated moment problem (<em>K</em>–RTMP) asks to characterize the existence of a positive Borel measure <em>μ</em>, supported on <em>K</em>, such that a linear functional <span><math><mi>L</mi></math></span>, defined on all rational functions of the form <span><math><mfrac><mrow><mi>f</mi></mrow><mrow><mi>q</mi></mrow></mfrac></math></span>, where <em>q</em> is a fixed polynomial with all real zeros of even order and <em>f</em> is any real polynomial of degree at most 2<em>k</em>, is an integration with respect to <em>μ</em>. The case of a compact set <em>K</em> was solved in <span><span>[4]</span></span>, but there is no argument that ensures that <em>μ</em> vanishes on all real zeros of <em>q</em>. An obvious necessary condition for the solvability of the <em>K</em>–RTMP is that <span><math><mi>L</mi></math></span> is nonnegative on every <em>f</em> satisfying <span><math><mi>f</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>K</mi></mrow></msub><mo>≥</mo><mn>0</mn></math></span>. If <span><math><mi>L</mi></math></span> is strictly positive on every <span><math><mn>0</mn><mo>≠</mo><mi>f</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>K</mi></mrow></msub><mo>≥</mo><mn>0</mn></math></span>, we add the missing argument from <span><span>[4]</span></span> and also bound the number of atoms in a minimal representing measure. We show by an example that nonnegativity of <span><math><mi>L</mi></math></span> is not sufficient and add the missing conditions to the solution. We also solve the <em>K</em>–RTMP for unbounded <em>K</em> and derive the solutions to the strong truncated Hamburger moment problem and the truncated moment problem on the unit circle as special cases.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 280-301"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164644","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":"On the complex stability radius for time-delay differential-algebraic systems","authors":"Alexander Malyshev , Miloud Sadkane","doi":"10.1016/j.laa.2024.12.007","DOIUrl":"10.1016/j.laa.2024.12.007","url":null,"abstract":"<div><div>An algorithm is proposed for computing the complex stability radius of a linear differential-algebraic system with a single delay and including a neutral term. The exponential factor in the characteristic equation is replaced by its Padé approximant thus reducing the level set method for finding the stability radius to a rational matrix eigenvalue problem. The level set method is coupled with a quadratically convergent iteration. An important condition relating the algebraic constraint and neutral term is introduced to eliminate the presence of characteristic roots approaching the imaginary axis at infinity. The number of iterations of the algorithm is roughly proportional to the numerical value of this condition. Effectiveness of the algorithm is illustrated by numerical examples.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 355-371"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164635","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":"The maximum spectral radius of P2k+1-free graphs of given size","authors":"Benju Wang, Bing Wang","doi":"10.1016/j.laa.2024.12.008","DOIUrl":"10.1016/j.laa.2024.12.008","url":null,"abstract":"<div><div>In this paper, we consider a Brualdi-Hoffman-Turán problem for graphs without path of given length. Denote by <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> the graph obtained from a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> by linking two vertices of distance two in the cycle. Recently, Li, Zhai and Shu showed that for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> and a graph <em>G</em> of size <span><math><mi>m</mi><mo>≥</mo><mn>16</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>, if <em>G</em> is <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free or <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-free, then the maximum adjacency spectral radius <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt><mo>)</mo></math></span>. It follows immediately that if <em>G</em> is <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free of size <span><math><mi>m</mi><mo>≥</mo><mn>16</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msqrt><mo>)</mo></math></span>. However, the upper bound is not sharp. We consider the case for <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graphs and obtain the following result: Let <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span> and <em>G</em> be a <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>-free graph of size <span><math><mi>m</mi><mo>≥</mo><mn>4</mn><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup></math></span>. Then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 302-314"},"PeriodicalIF":1.0,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164645","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":"Row or column completion of polynomial matrices of given degree II","authors":"Agurtzane Amparan , Itziar Baragaña , Silvia Marcaida , Alicia Roca","doi":"10.1016/j.laa.2024.12.004","DOIUrl":"10.1016/j.laa.2024.12.004","url":null,"abstract":"<div><div>The row (column) completion problem of polynomial matrices of given degree with prescribed eigenstructure has been studied in <span><span>[1]</span></span>, where several results of prescription of some of the four types of invariants that form the eigenstructure have also been obtained. In this paper we conclude the study, solving the completion for the cases not covered there. More precisely, we solve the row completion problem of a polynomial matrix when we prescribe the infinite (finite) structure and column and/or row minimal indices, and finally the column and/or row minimal indices. The necessity of the characterizations obtained holds to be true over arbitrary fields in all cases, whilst to prove the sufficiency it is required, in some of the cases, to work over algebraically closed fields.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 252-279"},"PeriodicalIF":1.0,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164643","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":"Some observations on Erdős matrices","authors":"Raghavendra Tripathi","doi":"10.1016/j.laa.2024.12.002","DOIUrl":"10.1016/j.laa.2024.12.002","url":null,"abstract":"<div><div>In a seminal paper in 1959, Marcus and Ree proved that every <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> bistochastic matrix <em>A</em> satisfies <span><math><msubsup><mrow><mo>‖</mo><mi>A</mi><mo>‖</mo></mrow><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≤</mo><msub><mrow><mi>max</mi></mrow><mrow><mi>σ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo></mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>σ</mi><mo>(</mo><mi>i</mi><mo>)</mo></mrow></msub></math></span> where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the symmetric group on <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Erdős asked to characterize the bistochastic matrices for which the equality holds in the Marcus–Ree inequality. We refer to such matrices as Erdős matrices. While this problem is trivial in dimension <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span>, the case of dimension <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> was only resolved recently in <span><span>[4]</span></span> in 2023. We prove that for every <em>n</em>, there are only finitely many <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Erdős matrices. We also give a complete characterization of Erdős matrices that yields an algorithm to generate all Erdős matrices in any given dimension. We also prove that Erdős matrices can have only rational entries. This answers a question of <span><span>[4]</span></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 236-251"},"PeriodicalIF":1.0,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164642","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":"Tridiagonal pairs of Krawtchouk type arising from finite-dimensional irreducible so4-modules","authors":"John Vincent S. Morales, Aaron Pagaygay","doi":"10.1016/j.laa.2024.12.001","DOIUrl":"10.1016/j.laa.2024.12.001","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be an algebraically closed field with <span><math><mi>char</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. The special linear algebra <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is the <span><math><mi>F</mi></math></span>-Lie algebra with Chevalley basis <span><math><mo>{</mo><mi>e</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>f</mi><mo>}</mo></math></span>. Since the special orthogonal algebra <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> is isomorphic to <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊕</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> is viewed as the <span><math><mi>F</mi></math></span>-Lie algebra with Chevalley basis <span><math><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></math></span>. In <span><span>[21, Lemma 3.1]</span></span>, there is an automorphism <span><math><mo>⁎</mo><mo>:</mo><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>→</mo><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> so that <span><math><mo>{</mo><msubsup><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>}</mo></math></span> is another Chevalley basis of <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. In <span><span>[21, Section 5]</span></span>, there is a simple construction of a finite-dimensional irreducible <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-module <em>V</em> on which <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> acts by derivation. In this paper, we construct four tridiagonal pairs (or TD pairs) on <em>V</em> via the action of the Chevalley bases of <span><math><msub><mrow><mi>so</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. We prove that these TD pairs are of Krawtchouk type and are not n","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 315-336"},"PeriodicalIF":1.0,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164637","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 Frobenius norm of the inverse of a non-negative matrix","authors":"Elsa Frankel , John Urschel","doi":"10.1016/j.laa.2024.11.030","DOIUrl":"10.1016/j.laa.2024.11.030","url":null,"abstract":"<div><div>We prove a new lower bound for the Frobenius norm of the inverse of an non-negative matrix. This bound is only a modest improvement over previous results, but is sufficient for fully resolving a conjecture of Harwitz and Sloane, commonly referred to as the S-matrix conjecture, for all dimensions larger than a small constant.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 193-203"},"PeriodicalIF":1.0,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165945","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 classes of graphs determined by the signless Laplacian spectrum","authors":"Jiachang Ye , Muhuo Liu , Zoran Stanić","doi":"10.1016/j.laa.2024.10.029","DOIUrl":"10.1016/j.laa.2024.10.029","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> denote the complete graph, the cycle and the path with <em>q</em> vertices, respectively. We use <span><math><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> to denote the signless Laplacian matrix of a simple undirected graph <em>G</em>, and say that <em>G</em> is determined by its signless Laplacian spectrum (for short, <em>G</em> is <em>DQS</em>) if there is no other non-isomorphic graph with the same signless Laplacian spectrum. In this paper, we prove the following results:<ul><li><span>(1)</span><span><div>If <span><math><mi>n</mi><mo>≥</mo><mn>21</mn></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, then <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∪</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is <em>DQS</em>;</div></span></li><li><span>(2)</span><span><div>If <span><math><mi>n</mi><mo>≥</mo><mn>21</mn></math></span> and <span><math><mn>3</mn><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>, then <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∪</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is <em>DQS</em> if and only if <span><math><mi>q</mi><mo>≠</mo><mn>3</mn></math></span>,</div></span></li></ul> where ∪ and ∨ stand for the disjoint union and the join of two graphs, respectively. Moreover, for <span><math><mi>q</mi><mo>=</mo><mn>3</mn></math></span> in <span><span>(2)</span></span> we identify <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>∪</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>5</mn><mo>)</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> as the unique graph sharing the signless Laplacian spectrum with the graph under consideration. Our results extend results of [Czechoslovak Math. J. 62 (2012) 1117–1134] and [Czechoslovak Math. J. 70 (2020) 21–31], where the authors showed that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>P</mi></mrow><mrow><mi","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 159-172"},"PeriodicalIF":1.0,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164621","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 complex symmetric weighted shifts","authors":"Chafiq Benhida , Piotr Budzyński","doi":"10.1016/j.laa.2024.11.031","DOIUrl":"10.1016/j.laa.2024.11.031","url":null,"abstract":"<div><div>Unbounded complex symmetric weighted shifts are studied. Complex symmetric unilateral weighted shifts whose <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> vectors contain the image of the canonical orthonormal basis under the conjugation are shown to be decomposable into an orthogonal sum of infinitely many complex selfadjoint truncated weighted shifts, which generalizes a result of S. Zhu and C.G. Li. The bilateral case is discussed as well. Additional results, examples, and open problems are supplied.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 217-235"},"PeriodicalIF":1.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164641","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}
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