Pub Date : 2024-07-25DOI: 10.1016/j.laa.2024.07.010
F.S. Benanti , O.M. Di Vincenzo , A. Valenti
Let F be an algebraically closed field of characteristic zero and G a cyclic group of odd prime order. We consider the class of finite dimensional -algebras, namely G-graded algebras endowed with graded involution ⁎, and we characterize the varieties generated by algebras of this class which are minimal with respect to the -exponent.
设 F 是特征为零的代数闭域和奇素数阶的循环群。我们考虑一类有限维-代数，即禀赋有分级反卷⁎的-分级代数，并描述由这一类代数生成的、关于-分量为最小的代数品种的特征。
{"title":"Minimal varieties of PI-algebras with graded involution","authors":"F.S. Benanti , O.M. Di Vincenzo , A. Valenti","doi":"10.1016/j.laa.2024.07.010","DOIUrl":"10.1016/j.laa.2024.07.010","url":null,"abstract":"<div><p>Let F be an algebraically closed field of characteristic zero and <em>G</em> a cyclic group of odd prime order. We consider the class of finite dimensional <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>⁎</mo><mo>)</mo></math></span>-algebras, namely <em>G</em>-graded algebras endowed with graded involution ⁎, and we characterize the varieties generated by algebras of this class which are minimal with respect to the <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mo>⁎</mo><mo>)</mo></math></span>-exponent.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003008/pdfft?md5=58c6ef0e78e02e4ca980c259967a3439&pid=1-s2.0-S0024379524003008-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873394","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}
Pub Date : 2024-07-24DOI: 10.1016/j.laa.2024.07.018
Alan Lew
<div><p>The <em>k</em>-th token graph of a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> is the graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> whose vertices are the <em>k</em>-subsets of <em>V</em> and whose edges are all pairs of <em>k</em>-subsets <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> such that the symmetric difference of <em>A</em> and <em>B</em> forms an edge in <em>G</em>. Let <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <em>G</em>, and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph <em>G</em> on <em>n</em> vertices and any <span><math><mn>0</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is contained in that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p><p>Here, we continue to study the relation between the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In particular, we show that, for <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, any eigenvalue <em>λ</em> of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> that is not contained in the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfies<span><span><span><math><mi>k</mi><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≤</mo><mi>λ</mi><mo>≤</mo><mi>k</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> is the second smallest eigenvalue of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (also known
{"title":"Garland's method for token graphs","authors":"Alan Lew","doi":"10.1016/j.laa.2024.07.018","DOIUrl":"10.1016/j.laa.2024.07.018","url":null,"abstract":"<div><p>The <em>k</em>-th token graph of a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> is the graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> whose vertices are the <em>k</em>-subsets of <em>V</em> and whose edges are all pairs of <em>k</em>-subsets <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> such that the symmetric difference of <em>A</em> and <em>B</em> forms an edge in <em>G</em>. Let <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <em>G</em>, and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph <em>G</em> on <em>n</em> vertices and any <span><math><mn>0</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is contained in that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p><p>Here, we continue to study the relation between the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In particular, we show that, for <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, any eigenvalue <em>λ</em> of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> that is not contained in the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfies<span><span><span><math><mi>k</mi><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≤</mo><mi>λ</mi><mo>≤</mo><mi>k</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> is the second smallest eigenvalue of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (also known ","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003082/pdfft?md5=bbdbf5062aa82eabe502f1effacd1b30&pid=1-s2.0-S0024379524003082-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872773","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}
Pub Date : 2024-07-24DOI: 10.1016/j.laa.2024.07.012
Erna Begović Kovač , Vjeran Hari
The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and J-Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators.
{"title":"Convergence of the complex block Jacobi methods under the generalized serial pivot strategies","authors":"Erna Begović Kovač , Vjeran Hari","doi":"10.1016/j.laa.2024.07.012","DOIUrl":"10.1016/j.laa.2024.07.012","url":null,"abstract":"<div><p>The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and <em>J</em>-Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872774","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}
Pub Date : 2024-07-22DOI: 10.1016/j.laa.2024.07.015
Alberto Daza-Garcia
On this work we study associative triple systems of the second kind. We show that for simple triple systems the automorphism group scheme is isomorphic to the automorphism group scheme of the 3-graded associative algebra with involution constructed by Loos. This result will allow us to prove our main result which is a complete classification up to isomorphism of the gradings of structurable algebras.
{"title":"Gradings on associative triple systems of the second kind","authors":"Alberto Daza-Garcia","doi":"10.1016/j.laa.2024.07.015","DOIUrl":"10.1016/j.laa.2024.07.015","url":null,"abstract":"<div><div>On this work we study associative triple systems of the second kind. We show that for simple triple systems the automorphism group scheme is isomorphic to the automorphism group scheme of the 3-graded associative algebra with involution constructed by Loos. This result will allow us to prove our main result which is a complete classification up to isomorphism of the gradings of structurable algebras.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785515","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}
Pub Date : 2024-07-19DOI: 10.1016/j.laa.2024.07.011
A. Guterman , E. Shen , I. Spitkovsky
The Gau-Wu number is an important matrix invariant describing the geometry of the numerical range. In this work, the group of non-singular linear preservers of the Gau-Wu number is completely characterized.
{"title":"On the group of linear preservers of the Gau-Wu number","authors":"A. Guterman , E. Shen , I. Spitkovsky","doi":"10.1016/j.laa.2024.07.011","DOIUrl":"10.1016/j.laa.2024.07.011","url":null,"abstract":"<div><p>The Gau-Wu number is an important matrix invariant describing the geometry of the numerical range. In this work, the group of non-singular linear preservers of the Gau-Wu number is completely characterized.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141778931","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}
Pub Date : 2024-07-18DOI: 10.1016/j.laa.2024.07.014
Aida Abiad , Bryan A. Curtis , Mary Flagg , H. Tracy Hall , Jephian C.-H. Lin , Bryan Shader
The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph G. In this paper, we refer to the i-nullity pair of a matrix A as , where is the matrix obtained from A by removing the i-th row and column. The inverse i-nullity pair problem is considered for complete graphs, cycles, and trees. The strong nullity interlacing property is introduced, and the corresponding supergraph lemma and decontraction lemma are developed as new tools for constructing matrices with a given nullity pair.
{"title":"The inverse nullity pair problem and the strong nullity interlacing property","authors":"Aida Abiad , Bryan A. Curtis , Mary Flagg , H. Tracy Hall , Jephian C.-H. Lin , Bryan Shader","doi":"10.1016/j.laa.2024.07.014","DOIUrl":"10.1016/j.laa.2024.07.014","url":null,"abstract":"<div><p>The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph <em>G</em>. In this paper, we refer to the <em>i</em>-nullity pair of a matrix <em>A</em> as <span><math><mo>(</mo><mi>null</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mi>null</mi><mo>(</mo><mi>A</mi><mo>(</mo><mi>i</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>A</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></span> is the matrix obtained from <em>A</em> by removing the <em>i</em>-th row and column. The inverse <em>i</em>-nullity pair problem is considered for complete graphs, cycles, and trees. The strong nullity interlacing property is introduced, and the corresponding supergraph lemma and decontraction lemma are developed as new tools for constructing matrices with a given nullity pair.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872808","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}
Pub Date : 2024-07-17DOI: 10.1016/j.laa.2024.07.013
Aina Mayumi , Gen Kimura , Hiromichi Ohno , Dariusz Chruściński
By employing a weighted Frobenius norm with a positive definite matrix ω, we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm and the standard Frobenius norm , there are exactly five possible generalizations, labeled (i) through (v), for the bounds on the norms of the commutator . In this paper, we establish the tight bounds for cases (iii) and (v), and propose conjectures regarding the tight bounds for cases (i) and (ii). Additionally, the tight bound for case (iv) is derived as a corollary of case (i). All these bounds (i)-(v) serve as generalizations of the BW inequality. The conjectured bounds for cases (i) and (ii) (and thus also (iv)) are numerically supported for matrices up to size . Proofs are provided for and certain special cases. Interestingly, we find applications of these bounds in quantum physics, particularly in the contexts of the uncertainty relation and open quantum dynamics.
{"title":"Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics","authors":"Aina Mayumi , Gen Kimura , Hiromichi Ohno , Dariusz Chruściński","doi":"10.1016/j.laa.2024.07.013","DOIUrl":"10.1016/j.laa.2024.07.013","url":null,"abstract":"<div><p>By employing a weighted Frobenius norm with a positive definite matrix <em>ω</em>, we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm <figure><img></figure> and the standard Frobenius norm <figure><img></figure>, there are exactly five possible generalizations, labeled (i) through (v), for the bounds on the norms of the commutator <span><math><mo>[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>]</mo><mo>:</mo><mo>=</mo><mi>A</mi><mi>B</mi><mo>−</mo><mi>B</mi><mi>A</mi></math></span>. In this paper, we establish the tight bounds for cases (iii) and (v), and propose conjectures regarding the tight bounds for cases (i) and (ii). Additionally, the tight bound for case (iv) is derived as a corollary of case (i). All these bounds (i)-(v) serve as generalizations of the BW inequality. The conjectured bounds for cases (i) and (ii) (and thus also (iv)) are numerically supported for matrices up to size <span><math><mi>n</mi><mo>=</mo><mn>15</mn></math></span>. Proofs are provided for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> and certain special cases. Interestingly, we find applications of these bounds in quantum physics, particularly in the contexts of the uncertainty relation and open quantum dynamics.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872807","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}
Pub Date : 2024-07-15DOI: 10.1016/j.laa.2024.07.007
A.H. Berliner , M. Catral , M. Cavers , S. Kim , P. van den Driessche
Given a pair of real symmetric matrices with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix . We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C.
给定一对实对称矩阵 A,B∈Rn×n，其非零图案由 n 个顶点上任意一对所选图形的边决定，我们考虑结构矩阵 C=[ABIO]∈R2n×2n的逆特征值问题。我们猜想，C 可以达到任何在共轭作用下封闭的谱。我们使用结构雅各布方法证明了阶最多为 4 或当 A 的图有一条汉密尔顿路径时的 A 和 B 的这一猜想，并证明了对 C 的特征值乘数有限制的任何一对图的这一猜想的较弱版本。
{"title":"An inverse eigenvalue problem for structured matrices determined by graph pairs","authors":"A.H. Berliner , M. Catral , M. Cavers , S. Kim , P. van den Driessche","doi":"10.1016/j.laa.2024.07.007","DOIUrl":"10.1016/j.laa.2024.07.007","url":null,"abstract":"<div><p>Given a pair of real symmetric matrices <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> with nonzero patterns determined by the edges of any pair of chosen graphs on <em>n</em> vertices, we consider an inverse eigenvalue problem for the structured matrix <span><math><mi>C</mi><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mspace></mspace><mi>A</mi><mspace></mspace></mtd><mtd><mi>B</mi></mtd></mtr><mtr><mtd><mspace></mspace><mi>I</mi><mspace></mspace></mtd><mtd><mi>O</mi></mtd></mtr></mtable><mo>]</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></mrow></msup></math></span>. We conjecture that <em>C</em> can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for <em>A</em> and <em>B</em> of orders at most 4 or when the graph of <em>A</em> has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of <em>C</em>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524002970/pdfft?md5=fa415a9b3c87c51014076976d43924d1&pid=1-s2.0-S0024379524002970-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141710709","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}
Pub Date : 2024-07-14DOI: 10.1016/j.laa.2024.07.006
Qian-Qian Chen , Ji-Ming Guo
A complex unit gain graph, or -gain graph, is a triple comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers , and a gain function with the property that . A cactus graph is a connected graph in which any two cycles have at most one vertex in common.
In this paper, we firstly show that there does not exist a complex unit gain graph with nullity , where , and are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30].
复数单位增益图或 T 增益图是一个三元组 Φ=(G,T,φ)，由作为 Φ 底图的简单图 G、单位复数集合 T={z∈C:|z|=1} 和增益函数 φ:E→→T 组成，其性质为 φ(eij)=φ(eji)-1 。本文首先证明不存在空性为 n(G)-2m(G)+2c(G)-1（其中 n(G)、m(G)和 c(G) 分别为 G 的阶、匹配数和循环数）的复数单位增益图。最后，我们描述了所有非星状复数单位增益双方形仙人掌图的特征，这概括了 Wong 等人（2022）[30] 的一个结果。
{"title":"Bounds of nullity for complex unit gain graphs","authors":"Qian-Qian Chen , Ji-Ming Guo","doi":"10.1016/j.laa.2024.07.006","DOIUrl":"10.1016/j.laa.2024.07.006","url":null,"abstract":"<div><p>A complex unit gain graph, or <span><math><mi>T</mi></math></span>-gain graph, is a triple <span><math><mi>Φ</mi><mo>=</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>φ</mi><mo>)</mo></math></span> comprised of a simple graph <em>G</em> as the underlying graph of Φ, the set of unit complex numbers <span><math><mi>T</mi><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>C</mi><mo>:</mo><mo>|</mo><mi>z</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span>, and a gain function <span><math><mi>φ</mi><mo>:</mo><mover><mrow><mi>E</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>→</mo><mi>T</mi></math></span> with the property that <span><math><mi>φ</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>φ</mi><msup><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. A cactus graph is a connected graph in which any two cycles have at most one vertex in common.</p><p>In this paper, we firstly show that there does not exist a complex unit gain graph with nullity <span><math><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>2</mn><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are the order, matching number, and cyclomatic number of <em>G</em>. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) <span><span>[30]</span></span>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141701119","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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