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}
{"title":"Numerical range of real-valued linear mapping on the complex Stiefel manifold: Convexity and application","authors":"Hanzhi Chen, Zhenhong Huang, Mengmeng Song, Yong Xia","doi":"10.1016/j.laa.2025.01.031","DOIUrl":"10.1016/j.laa.2025.01.031","url":null,"abstract":"<div><div>The study confirms the convexity of the joint numerical range of any <em>k</em> real-valued linear functions on the <span><math><mi>n</mi><mo>×</mo><mi>p</mi></math></span> complex Stiefel manifold under the condition <span><math><mi>k</mi><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>1</mn></math></span>. Revealing the hidden convexity of fractional linear programming on the complex Stiefel manifold, a first-time study, serves as an impactful application.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 95-110"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138083","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 graphs for disjoint odd wheels","authors":"Yu Luo , Zhenyu Ni , Yanxia Dong","doi":"10.1016/j.laa.2025.01.034","DOIUrl":"10.1016/j.laa.2025.01.034","url":null,"abstract":"<div><div>For a given graph <em>F</em>, let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all <em>F</em>-free graphs of order <em>n</em>, respectively. <span><math><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> consist of the extremal graphs associated with <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> and <span><math><mrow><mi>spex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, respectively. The odd wheel <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is constructed by joining a vertex to a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub></math></span>. Cioabă, Desai and Tait determined the spectral extremal graphs of <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>k</mi><mo>∉</mo><mrow><mo>{</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></mrow></math></span>. Xiao and Zamora determined the Turán number and all extremal graphs for <span><math><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, which is the union of <em>t</em> vertex-disjoint copies of <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>. In this paper, we focus on the graph with maximum spectral radius among those that exclude any subgraph isomorphic to <span><math><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. We present structural characteristics of these spectral extremal graphs for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn><mo>,</mo><mi>k</mi><mo>∉</mo><mrow><mo>{</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></mrow></math></span>. Furthermore, we demonstrate that <span><math><mrow><mi>SPEX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>∩</mo><mrow><mi>EX</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>t</mi><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> for <span><math><mi>k</mi><mo>≥</mo><mn>10</mn></math></span> and <em>n</em> large enough.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 243-266"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348007","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":"Constructing adjacency and distance cospectral graphs via regular rational orthogonal matrix","authors":"Lihuan Mao , Yuanhang Xu , Fenjin Liu , Bei Liu","doi":"10.1016/j.laa.2025.01.033","DOIUrl":"10.1016/j.laa.2025.01.033","url":null,"abstract":"<div><div>Two graphs <em>G</em> and <em>H</em> are <em>cospectral</em> if they share the same spectrum. Constructing <em>cospectral</em> non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature. In this paper, we construct infinite families of adjacency cospectral graphs through the GM-switching method based on generalized Johnson graphs. We give some graph operations (e.g. rooted-product, corona, cartesian product, and coalescence) to construct distance cospectral graphs with different edges via a regular rational orthogonal matrix.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 111-128"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138082","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":"Polynomial codimension growth of superalgebras with superautomorphism","authors":"Sara Accomando","doi":"10.1016/j.laa.2025.01.035","DOIUrl":"10.1016/j.laa.2025.01.035","url":null,"abstract":"<div><div>In this paper we present some results concerning associative superalgebras endowed with a superautomorphism of order ≤2. We characterize the superalgebras with superautomorphism with multiplicities of the cocharacter bounded by a constant. Moreover, we determine the characterization of the superalgebras with superautomorphism with polynomial growth of the codimensions and we give a classification of the subvarieties of the varieties of almost polynomial growth. Finally, we characterize the superalgebras with superautomorphism with linear growth of the codimensions.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 50-79"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143137954","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}
S.P. Glasby , Alice C. Niemeyer , Cheryl E. Praeger
{"title":"Bipartite q-Kneser graphs and two-generated irreducible linear groups","authors":"S.P. Glasby , Alice C. Niemeyer , Cheryl E. Praeger","doi":"10.1016/j.laa.2025.01.032","DOIUrl":"10.1016/j.laa.2025.01.032","url":null,"abstract":"<div><div>Let <span><math><mi>V</mi><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span> be a <em>d</em>-dimensional vector space over the field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of order <em>q</em>. Fix positive integers <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> satisfying <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mi>d</mi></math></span>. Motivated by analysing a fundamental algorithm in computational group theory for recognising classical groups, we consider a certain quantity <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> which arises in both graph theory and group representation theory: <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> is the proportion of 3-walks in the ‘bipartite <em>q</em>-Kneser graph’ <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> that are closed 3-arcs. We prove that, for a group <em>G</em> satisfying <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>⊴</mo><mi>G</mi><mo>⩽</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, the proportion of certain element-pairs in <em>G</em> called ‘<span><math><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-stingray duos’ which generate an irreducible subgroup is also equal to <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. We give an exact formula for <span><math><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, and prove that<span><span><span><math><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo><</mo><mi>P</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo><</mo><mn>1</mn","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 203-229"},"PeriodicalIF":1.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387463","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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