{"title":"On a question of Bourin","authors":"Fuad Kittaneh , Éric Ricard","doi":"10.1016/j.laa.2025.02.004","DOIUrl":"10.1016/j.laa.2025.02.004","url":null,"abstract":"<div><div>We use complex analysis methods to give a partial answer to a question by Bourin in matrix inequalities.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 356-362"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378779","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 commutators of unipotent matrices of index 2","authors":"Kennett L. Dela Rosa, Juan Paolo C. Santos","doi":"10.1016/j.laa.2025.02.003","DOIUrl":"10.1016/j.laa.2025.02.003","url":null,"abstract":"<div><div>A commutator of unipotent matrices of index 2 is a matrix of the form <span><math><mi>X</mi><mi>Y</mi><msup><mrow><mi>X</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>Y</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>, where <em>X</em> and <em>Y</em> are unipotent matrices of index 2, that is, <span><math><mi>X</mi><mo>≠</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>Y</mi><mo>≠</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and <span><math><msup><mrow><mo>(</mo><mi>X</mi><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>Y</mi><mo>−</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msub><mrow><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msub></math></span>. If <span><math><mi>n</mi><mo>></mo><mn>2</mn></math></span> and <span><math><mi>F</mi></math></span> is a field with <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math></span>, then it is shown that every <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix over <span><math><mi>F</mi></math></span> with determinant 1 is a product of at most four commutators of unipotent matrices of index 2. Consequently, every <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix over <span><math><mi>F</mi></math></span> with determinant 1 is a product of at most eight unipotent matrices of index 2. Conditions on <span><math><mi>F</mi></math></span> are given that improve the upper bound on the commutator factors from four to three or two. The situation for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> is also considered. This study reveals a connection between factorability into commutators of unipotent matrices and properties of <span><math><mi>F</mi></math></span> such as its characteristic or its set of perfect squares.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 385-404"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394763","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":"Spectral extremal problems for graphs with bounded clique number","authors":"Tingting Wang, Lihua Feng, Lu Lu","doi":"10.1016/j.laa.2025.01.043","DOIUrl":"10.1016/j.laa.2025.01.043","url":null,"abstract":"<div><div>For a family of graphs <span><math><mi>F</mi></math></span>, a graph is called <span><math><mi>F</mi></math></span>-free if it contains no subgraph isomorphic to any graph in <span><math><mi>F</mi></math></span>. For two integers <em>n</em> and <em>r</em>, let <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> be the set of graphs on <em>n</em> vertices with clique number at most <em>r</em>. Denote by<span><span><span><math><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>G</mi><mspace></mspace><mrow><mtext>is </mtext><mi>F</mi><mtext>-free</mtext></mrow><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the spectral radius of <em>G</em>. Furthermore, denote by <span><math><msub><mrow><mtext>Ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>G</mi><mo>∈</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>|</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mtext>ex</mtext></mrow><mrow><mi>s</mi><mi>p</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>G</mi><mtext> is </mtext><mi>F</mi><mtext>-</mtext><mtext>free</mtext><mo>}</mo></math></span> the set of extremal graphs. In this paper, we first give a spectral Erdös-Sós theorem in <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>, that is, for fixed <span><math><mi>k</mi><mo>≥</mo><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> and sufficiently large <em>n</em>, if a graph <span><math><mi>G</mi><mo>∈</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> with <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>λ</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>∨</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo>)</mo></math></span>, then it contains all trees on <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn></math></span> vertices or <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>3</mn></math></span> vertices unless <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo><mo>∨</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></math></span>, where <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>k</mi><mo","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 273-295"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143377585","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}
Nicholas J. Higham , Matthew C. Lettington , Karl Michael Schmidt
{"title":"Symmetry decomposition and matrix multiplication","authors":"Nicholas J. Higham , Matthew C. Lettington , Karl Michael Schmidt","doi":"10.1016/j.laa.2025.01.041","DOIUrl":"10.1016/j.laa.2025.01.041","url":null,"abstract":"<div><div>General matrices can be split uniquely into Frobenius-orthogonal components: a constant row and column sum (type S) part, a vertex cross sum (type V) part and a weight part. We show that for square matrices, the type S part can be expressed as a sum of squares of type V matrices. We investigate the properties of such decomposition under matrix multiplication, in particular how the pseudoinverses of a matrix relate to the pseudoinverses of its component parts. For invertible matrices, this yields an expression for the inverse where only the type S part needs to be (pseudo)inverted; in the example of the Wilson matrix, this component is considerably better conditioned than the whole matrix. We also show a relation between matrix determinants and the weight of their matrix inverses and give a simple proof for Frobenius-optimal approximations with the constant row and column sum and the vertex cross sum properties, respectively, to a given matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 310-335"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378906","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":"Additive mappings preserving orthogonality between complex inner product spaces","authors":"Lei Li , Siyu Liu , Antonio M. Peralta","doi":"10.1016/j.laa.2025.01.042","DOIUrl":"10.1016/j.laa.2025.01.042","url":null,"abstract":"<div><div>Let <em>H</em> and <em>K</em> be two complex inner product spaces with dim<span><math><mo>(</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mn>2</mn></math></span>. We prove that for each non-zero mapping <span><math><mi>A</mi><mo>:</mo><mi>H</mi><mo>→</mo><mi>K</mi></math></span> with dense image the following statements are equivalent:<ul><li><span>(<em>a</em>)</span><span><div><em>A</em> is (complex) linear or conjugate-linear mapping and there exists <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span> such that <span><math><mo>‖</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>‖</mo><mo>=</mo><mi>γ</mi><mo>‖</mo><mi>x</mi><mo>‖</mo></math></span>, for all <span><math><mi>x</mi><mo>∈</mo><mi>H</mi></math></span>, that is, <em>A</em> is a positive scalar multiple of a linear or a conjugate-linear isometry;</div></span></li><li><span>(<em>a</em>)</span><span><div>There exists <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>></mo><mn>0</mn></math></span> such that one of the next properties holds for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>H</mi></math></span>:<ul><li><span><span><math><mo>(</mo><mi>b</mi><mo>.</mo><mn>1</mn><mo>)</mo></math></span></span><span><div><span><math><mo>〈</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>〉</mo><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>〈</mo><mi>x</mi><mo>|</mo><mi>y</mi><mo>〉</mo></math></span>,</div></span></li><li><span><span><math><mo>(</mo><mi>b</mi><mo>.</mo><mn>1</mn><mo>)</mo></math></span></span><span><div><span><math><mo>〈</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>〉</mo><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>〈</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>〉</mo></math></span>;</div></span></li></ul></div></span></li><li><span>(<em>a</em>)</span><span><div><em>A</em> is linear or conjugate-linear and preserves orthogonality;</div></span></li><li><span>(<em>a</em>)</span><span><div><em>A</em> is additive and preserves orthogonality in both directions;</div></span></li><li><span>(<em>a</em>)</span><span><div><em>A</em> is additive and preserves orthogonality.</div></span></li></ul> This extends to the complex setting a recent generalization of the Koldobsky–Blanco–Turnšek theorem obtained by Wójcik for real normed spaces.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 448-457"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394634","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}
Hassane Benbouziane, Kaddour Chadli, Mustapha Ech-chérif El Kettani
{"title":"Mappings preserving generalized and hyper-generalized projection operators","authors":"Hassane Benbouziane, Kaddour Chadli, Mustapha Ech-chérif El Kettani","doi":"10.1016/j.laa.2025.01.038","DOIUrl":"10.1016/j.laa.2025.01.038","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 on a complex Hilbert space <span><math><mi>H</mi></math></span> with <span><math><mi>dim</mi><mspace></mspace><mi>H</mi><mo>≥</mo><mn>3</mn></math></span>. For a fixed integer <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, an operator <span><math><mi>A</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is called <em>k</em>-generalized projection if <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, and <em>k</em>-hyper-generalized projection if <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span>, where <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> and <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> denote the adjoint and the Moore–Penrose inverse of <em>A</em>, respectively. In this paper, we provide a complete characterization of surjective maps <span><math><mi>Φ</mi><mo>:</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>→</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> such that <span><math><mi>A</mi><mo>−</mo><mi>λ</mi><mi>B</mi></math></span> is <em>k</em>-generalized projection (resp. <em>k</em>-hyper-generalized projection) if and only if <span><math><mi>Φ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>−</mo><mi>λ</mi><mi>Φ</mi><mo>(</mo><mi>B</mi><mo>)</mo></math></span> is <em>k</em>-generalized projection (resp. <em>k</em>-hyper-generalized projection), for any <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><mi>λ</mi><mo>∈</mo><mi>C</mi></math></span>. We also study the non-linear preservers of <em>k</em>-potent operators.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 418-447"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394830","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":"Further restricted versions of the nonnegative matrix factorization problem","authors":"Yaroslav Shitov","doi":"10.1016/j.laa.2025.01.040","DOIUrl":"10.1016/j.laa.2025.01.040","url":null,"abstract":"<div><div>We discuss the functions of SNT-rank and restricted SNT-rank, introduced in a recent article of Kokol Bukovšek and Šmigoc. We answer several questions from their work and give an example of a symmetric nonnegative matrix for which the restricted SNT-rank is not defined. Moreover, we show that the restricted SNT-rank of a matrix can exceed its SNT-rank even if both of them are defined. We use earlier results to give bounds on SNT-ranks of rank-three matrices and Euclidean distance matrices, and we determine the complexity of the algorithmic computation of SNT-ranks.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 267-272"},"PeriodicalIF":1.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143360790","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":"Jordan type stratification of spaces of commuting nilpotent matrices","authors":"Mats Boij , Anthony Iarrobino , Leila Khatami","doi":"10.1016/j.laa.2025.01.039","DOIUrl":"10.1016/j.laa.2025.01.039","url":null,"abstract":"<div><div>An <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> nilpotent matrix <em>B</em> is determined up to conjugacy by a partition <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>B</mi></mrow></msub></math></span> of <em>n</em>, its <em>Jordan type</em> given by the sizes of its Jordan blocks. The Jordan type <span><math><mi>D</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> of a nilpotent matrix in the dense orbit of the nilpotent commutator of a given nilpotent matrix of Jordan type <em>P</em> is <em>stable</em> - has parts differing pairwise by at least two - and was determined by R. Basili. The second two authors, with B. Van Steirteghem and R. Zhao determined a rectangular table of partitions <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> having a given stable partition <em>Q</em> as the Jordan type of its maximum nilpotent commutator. They proposed a box conjecture, that would generalize the answer to stable partitions <em>Q</em> having <em>ℓ</em> parts: it was proven recently by J. Irving, T. Košir and M. Mastnak.</div><div>Using this result and also some tropical calculations, the authors here determine equations defining the loci of each partition in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>Q</mi><mo>)</mo></math></span>, when <em>Q</em> is stable with two parts. The equations for each locus form a complete intersection. The authors propose a conjecture generalizing their result to arbitrary stable <em>Q</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 183-202"},"PeriodicalIF":1.0,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277787","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":"A note on the Bollobás-Nikiforov conjecture","authors":"Jiasheng Zeng, Xiao-Dong Zhang","doi":"10.1016/j.laa.2025.01.037","DOIUrl":"10.1016/j.laa.2025.01.037","url":null,"abstract":"<div><div>Bollobás and Nikiforov <span><span>[2]</span></span> proposed a conjecture that for any non-complete graph <em>G</em> with <em>m</em> edges and clique number <em>ω</em>, the following inequality holds:<span><span><span><math><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≤</mo><mn>2</mn><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ω</mi></mrow></mfrac><mo>)</mo></mrow><mi>m</mi><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are the two largest eigenvalues of the adjacency matrix <span><math><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span>. Later, Elphick, Linz, and Wocjan <span><span>[6]</span></span> proposed a generalization of this conjecture. In this paper, we prove that the conjecture proposed by Bollobás and Nikiforov holds for both line graphs and graphs with at most <span><math><mfrac><mrow><msqrt><mrow><mn>6</mn></mrow></msqrt></mrow><mrow><mn>27</mn></mrow></mfrac><msup><mrow><mi>m</mi></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> triangles, and that the generalized conjecture holds for both line graphs with additional conditions and graphs with not many triangles, which extends and strengthens some known results.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 230-242"},"PeriodicalIF":1.0,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277785","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":"Hoffman colorings of graphs","authors":"Aida Abiad , Wieb Bosma , Thijs van Veluw","doi":"10.1016/j.laa.2025.01.036","DOIUrl":"10.1016/j.laa.2025.01.036","url":null,"abstract":"<div><div>Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for general graphs not much is known. We investigate tightness of the Hoffman bound, with a particular focus on irregular graphs, obtaining several results on the graph structure of Hoffman colorings. In particular, we prove a Decomposition Theorem, which characterizes the structure of Hoffman colorings, and we use it to completely classify Hoffman colorability of cone graphs and line graphs. We also prove a partial converse, the Composition Theorem, leading to an algorithm for computing all connected Hoffman colorable graphs for some given number of vertices and colors. Since several graph coloring parameters are known to be sandwiched between the Hoffman bound and the chromatic number, as a byproduct of our results, we obtain the values of these chromatic parameters.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 129-150"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138085","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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