Pub Date : 2024-11-07DOI: 10.1016/j.laa.2024.11.005
Rounak Biswas, Falguni Roy
For two idempotent matrix , let alg denote the smallest subalgebra of that contains and the identity matrix . This paper provides a complete classification of alg without imposing any restrictions on P and Q. As a result of this classification, the issue of group invertibility within alg is fully resolved.
对于两个幂等矩阵 P,Q∈Cn×n, 让 alg(In,P,Q) 表示 Cn×n 中包含 P,Q 和同一矩阵 In 的最小子代数。本文在不对 P 和 Q 施加任何限制的情况下，对 alg(In,P,Q) 进行了完整的分类。
{"title":"Comprehensive classification of the algebra generated by two idempotent matrices","authors":"Rounak Biswas, Falguni Roy","doi":"10.1016/j.laa.2024.11.005","DOIUrl":"10.1016/j.laa.2024.11.005","url":null,"abstract":"<div><div>For two idempotent matrix <span><math><mi>P</mi><mo>,</mo><mi>Q</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, let alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> denote the smallest subalgebra of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span> that contains <span><math><mi>P</mi><mo>,</mo><mi>Q</mi></math></span> and the identity matrix <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This paper provides a complete classification of alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> without imposing any restrictions on <em>P</em> and <em>Q</em>. As a result of this classification, the issue of group invertibility within alg<span><math><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> is fully resolved.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 185-206"},"PeriodicalIF":1.0,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653347","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 : 2024-11-07DOI: 10.1016/j.laa.2024.11.002
Francesca Albertini , Domenico D'Alessandro
In the analysis of controllability of finite dimensional quantum systems, subspace controllability refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics.
We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state (Pure State Controllability) is verified.
{"title":"Quantum subspace controllability implying full controllability","authors":"Francesca Albertini , Domenico D'Alessandro","doi":"10.1016/j.laa.2024.11.002","DOIUrl":"10.1016/j.laa.2024.11.002","url":null,"abstract":"<div><div>In the analysis of controllability of finite dimensional quantum systems, <em>subspace controllability</em> refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics.</div><div>We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state (<em>Pure State Controllability</em>) is verified.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 207-229"},"PeriodicalIF":1.0,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653348","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 : 2024-11-06DOI: 10.1016/j.laa.2024.11.004
Mario Angelelli
We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in d indeterminates, whose exponents define a -valued set function. By combining the Grassmann-Plücker relations for the two matrices, we derive a family of sparse polynomials, whose factorisation properties in a Laurent polynomial ring are studied and related to information-theoretic notions.
Under a given genericity condition, we prove the equivalence between combinatorial reductions and determinantal expansions with invertible minor products; specifically, a deformation returns a determinantal expansion if and only if it is induced by a diagonal matrix of units in acting as a kernel in the original determinant expression. This characterisation supports the definition of a new method for checking and recovering combinatorial reductions for matroid permutations.
我们以两个矩阵乘积的行列式展开为基础，引入一个代数模型来检验集合函数的组合还原。行列式展开的每个项都通过 d 个不定项的单项式因子变形，其指数定义了一个 Zd 值集合函数。通过结合两个矩阵的格拉斯曼-普吕克关系，我们推导出了稀疏多项式族，研究了它们在劳伦多项式环中的因式分解性质，并将其与信息论概念联系起来。在给定的通性条件下，我们证明了组合还原和行列式展开之间的等价性；具体地说，当且仅当变形是由 C(t) 中作为原始行列式表达式核的单位对角矩阵诱导时，它才返回行列式展开。这一特征支持定义一种新方法，用于检查和恢复矩阵排列的组合还原。
{"title":"Combinatorial reduction of set functions and matroid permutations through minor invertible product assignment","authors":"Mario Angelelli","doi":"10.1016/j.laa.2024.11.004","DOIUrl":"10.1016/j.laa.2024.11.004","url":null,"abstract":"<div><div>We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in <em>d</em> indeterminates, whose exponents define a <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-valued set function. By combining the Grassmann-Plücker relations for the two matrices, we derive a family of sparse polynomials, whose factorisation properties in a Laurent polynomial ring are studied and related to information-theoretic notions.</div><div>Under a given genericity condition, we prove the equivalence between combinatorial reductions and determinantal expansions with invertible minor products; specifically, a deformation returns a determinantal expansion if and only if it is induced by a diagonal matrix of units in <span><math><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> acting as a kernel in the original determinant expression. This characterisation supports the definition of a new method for checking and recovering combinatorial reductions for matroid permutations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 89-128"},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653344","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 : 2024-11-06DOI: 10.1016/j.laa.2024.11.001
Alec Dektor
We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.
{"title":"Collocation methods for nonlinear differential equations on low-rank manifolds","authors":"Alec Dektor","doi":"10.1016/j.laa.2024.11.001","DOIUrl":"10.1016/j.laa.2024.11.001","url":null,"abstract":"<div><div>We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 143-184"},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653346","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 : 2024-11-06DOI: 10.1016/j.laa.2024.10.028
M. Cavers , B. Miraftab
This paper provides insight into the problem of characterizing digraphs (with loops permitted) that have few distinct adjacency eigenvalues, or equivalently, characterizing square -matrices that have few distinct eigenvalues. A spectral characterization of strongly connected digraphs whose adjacency matrix has exactly two distinct eigenvalues is given and constructions of such digraphs are described. In addition, bipartite digraphs with exactly three distinct eigenvalues are discussed.
{"title":"Digraphs with few distinct eigenvalues","authors":"M. Cavers , B. Miraftab","doi":"10.1016/j.laa.2024.10.028","DOIUrl":"10.1016/j.laa.2024.10.028","url":null,"abstract":"<div><div>This paper provides insight into the problem of characterizing digraphs (with loops permitted) that have few distinct adjacency eigenvalues, or equivalently, characterizing square <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-matrices that have few distinct eigenvalues. A spectral characterization of strongly connected digraphs whose adjacency matrix has exactly two distinct eigenvalues is given and constructions of such digraphs are described. In addition, bipartite digraphs with exactly three distinct eigenvalues are discussed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 129-142"},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653345","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 : 2024-10-31DOI: 10.1016/j.laa.2024.10.027
Tian-Xiao He
In this paper, we define double almost-Riordan arrays and find that the set of all double almost-Riordan arrays forms a group, called the double almost-Riordan group. We also obtain the sequence characteristics of double almost-Riordan arrays and give the production matrices of two types for double almost-Riordan arrays. In addition, we discuss the algebraic properties of the double almost-Riordan group, and finally give the compression of double almost-Riordan arrays and their sequence characteristics.
{"title":"The double almost-Riordan group","authors":"Tian-Xiao He","doi":"10.1016/j.laa.2024.10.027","DOIUrl":"10.1016/j.laa.2024.10.027","url":null,"abstract":"<div><div>In this paper, we define double almost-Riordan arrays and find that the set of all double almost-Riordan arrays forms a group, called the double almost-Riordan group. We also obtain the sequence characteristics of double almost-Riordan arrays and give the production matrices of two types for double almost-Riordan arrays. In addition, we discuss the algebraic properties of the double almost-Riordan group, and finally give the compression of double almost-Riordan arrays and their sequence characteristics.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 50-88"},"PeriodicalIF":1.0,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653401","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 : 2024-10-31DOI: 10.1016/j.laa.2024.10.026
A.Y. Alfakih
The Colin de Verdière number of graph G, denoted by , is a spectral invariant of G that is related to some of its topological properties. For example, iff G is planar. A penny graph is the contact graph of equal-radii disks with disjoint interiors in the plane. In this note, we prove lower bounds on when the complement is a penny graph.
图 G 的 Colin de Verdière 数（用 μ(G)表示）是图 G 的谱不变量，与图 G 的某些拓扑特性有关。例如，如果 G 是平面图，μ(G)≤3。一分钱图是平面上内部相交的等辐圆盘的接触图。在本论文中，我们将证明当补集 G‾ 是便士图时，μ(G) 的下界。
{"title":"On the Colin de Verdière graph number and penny graphs","authors":"A.Y. Alfakih","doi":"10.1016/j.laa.2024.10.026","DOIUrl":"10.1016/j.laa.2024.10.026","url":null,"abstract":"<div><div>The Colin de Verdière number of graph <em>G</em>, denoted by <span><math><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is a spectral invariant of <em>G</em> that is related to some of its topological properties. For example, <span><math><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>3</mn></math></span> iff <em>G</em> is planar. A <em>penny graph</em> is the contact graph of equal-radii disks with disjoint interiors in the plane. In this note, we prove lower bounds on <span><math><mi>μ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> when the complement <span><math><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover></math></span> is a penny graph.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 17-25"},"PeriodicalIF":1.0,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592670","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 : 2024-10-30DOI: 10.1016/j.laa.2024.10.025
Gregor Dolinar , Dijana Ilišević , Bojan Kuzma , Janko Marovt
Let be the cone of all positive semidefinite real matrices. We describe the form of all surjective maps on , , that preserve the minus partial order in both directions.
{"title":"Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order","authors":"Gregor Dolinar , Dijana Ilišević , Bojan Kuzma , Janko Marovt","doi":"10.1016/j.laa.2024.10.025","DOIUrl":"10.1016/j.laa.2024.10.025","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> be the cone of all positive semidefinite <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> real matrices. We describe the form of all surjective maps on <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, that preserve the minus partial order in both directions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 26-49"},"PeriodicalIF":1.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653400","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 : 2024-10-28DOI: 10.1016/j.laa.2024.10.023
Ao Fan , Ruifang Liu , Guoyan Ao
The binding number of a graph G is the minimum value of taken over all non-empty subsets X of such that . A graph G is called 1-binding if . Let b be a positive integer. An odd-factor of a graph G is a spanning subgraph F such that for each , is odd and . Motivated by the result of Fan, Lin and Lu (2022) [10] on the existence of an odd -factor in connected graphs, we first present a tight sufficient condition in terms of the spectral radius for a connected 1-binding graph to contain an odd -factor, which generalizes the result of Fan and Lin (2024) [8] on the existence of a 1-factor in 1-binding graphs.
A spanning k-tree is a spanning tree with the degree of every vertex at most k, which is considered as a connected -factor. Inspired by the result of Fan, Goryainov, Huang and Lin (2022) [9] on the existence of a spanning k-tree in connected graphs, we in this paper provide a tight sufficient condition based on the spectral radius for a connected 1-binding graph to contain a spanning k-tree.
图 G 的绑定数 b(G) 是在 V(G) 的所有非空子集 X 上取的|NG(X)|/|X|的最小值，使得 NG(X)≠V(G) 。如果 b(G)≥1，则称图形 G 为 1 约束。图 G 的奇数 [1,b]- 因子是一个跨子图 F，对于每个 v∈V(G)，dF(v) 都是奇数且 1≤dF(v)≤b 。受 Fan、Lin 和 Lu (2022) [10] 关于连通图中存在奇数 [1,b]- 因子的结果的启发，我们首先从谱半径的角度提出了连通的 1 绑定图包含奇数 [1,b]- 因子的严密充分条件，它概括了 Fan 和 Lin (2024) [8] 关于 1 绑定图中存在 1 因子的结果。生成 k 树是指每个顶点的度最多为 k 的生成树，它被视为连通的 [1,k]- 因子。受 Fan、Goryainov、Huang 和 Lin（2022）[9] 关于连通图中存在生成 k 树的结果的启发，我们在本文中提供了一个基于谱半径的连通 1 约束图包含生成 k 树的严密充分条件。
{"title":"Spectral radius, odd [1,b]-factor and spanning k-tree of 1-binding graphs","authors":"Ao Fan , Ruifang Liu , Guoyan Ao","doi":"10.1016/j.laa.2024.10.023","DOIUrl":"10.1016/j.laa.2024.10.023","url":null,"abstract":"<div><div>The <em>binding number</em> <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the minimum value of <span><math><mo>|</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>|</mo><mo>/</mo><mo>|</mo><mi>X</mi><mo>|</mo></math></span> taken over all non-empty subsets <em>X</em> of <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>≠</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. A graph <em>G</em> is called 1<em>-binding</em> if <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>1</mn></math></span>. Let <em>b</em> be a positive integer. An <em>odd</em> <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span><em>-factor</em> of a graph <em>G</em> is a spanning subgraph <em>F</em> such that for each <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> is odd and <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>b</mi></math></span>. Motivated by the result of Fan, Lin and Lu (2022) <span><span>[10]</span></span> on the existence of an odd <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor in connected graphs, we first present a tight sufficient condition in terms of the spectral radius for a connected 1-binding graph to contain an odd <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>-factor, which generalizes the result of Fan and Lin (2024) <span><span>[8]</span></span> on the existence of a 1-factor in 1-binding graphs.</div><div>A spanning <em>k</em>-tree is a spanning tree with the degree of every vertex at most <em>k</em>, which is considered as a connected <span><math><mo>[</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>]</mo></math></span>-factor. Inspired by the result of Fan, Goryainov, Huang and Lin (2022) <span><span>[9]</span></span> on the existence of a spanning <em>k</em>-tree in connected graphs, we in this paper provide a tight sufficient condition based on the spectral radius for a connected 1-binding graph to contain a spanning <em>k</em>-tree.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"705 ","pages":"Pages 1-16"},"PeriodicalIF":1.0,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586752","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 : 2024-10-22DOI: 10.1016/j.laa.2024.10.021
Nahid Gharakhanlu , Mohammad Sal Moslehian
We establish the subadditivity of strongly operator convex functions on and . By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on . We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on . Moreover, we study strongly operator convex functions on with and on the left half-line with . We demonstrate that any nonconstant strongly operator convex function on is strictly operator decreasing, and any nonconstant strongly operator convex function on is strictly operator monotone. Consequently, for a strongly operator convex function g on or , we provide lower bounds for whenever .
{"title":"Around strongly operator convex functions","authors":"Nahid Gharakhanlu , Mohammad Sal Moslehian","doi":"10.1016/j.laa.2024.10.021","DOIUrl":"10.1016/j.laa.2024.10.021","url":null,"abstract":"<div><div>We establish the subadditivity of strongly operator convex functions on <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>. Moreover, we study strongly operator convex functions on <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> with <span><math><mo>−</mo><mo>∞</mo><mo><</mo><mi>a</mi></math></span> and on the left half-line <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>b</mi><mo>)</mo></math></span> with <span><math><mi>b</mi><mo><</mo><mo>∞</mo></math></span>. We demonstrate that any nonconstant strongly operator convex function on <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> is strictly operator decreasing, and any nonconstant strongly operator convex function on <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>b</mi><mo>)</mo></math></span> is strictly operator monotone. Consequently, for a strongly operator convex function <em>g</em> on <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> or <span><math><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, we provide lower bounds for <span><math><mo>|</mo><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>−</mo><mi>g</mi><mo>(</mo><mi>B</mi><mo>)</mo><mo>|</mo></math></span> whenever <span><math><mi>A</mi><mo>−</mo><mi>B</mi><mo>></mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 231-248"},"PeriodicalIF":1.0,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554408","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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