Linear Algebra and its Applications最新文献

{"title":"Generating functions of non-backtracking walks on weighted digraphs: Radius of convergence and Ihara's theorem","authors":"Vanni Noferini ,&nbsp;María C. Quintana","doi":"10.1016/j.laa.2024.06.022","DOIUrl":"https://doi.org/10.1016/j.laa.2024.06.022","url":null,"abstract":"<div><p>It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In Grindrod et al. <span>[13]</span>, the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed (unweighted or weighted) graphs, showing that it depends on the number of cycles in the undirectization of the graph. We also consider backtrack-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case. Finally, for weighted directed graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound, and we also prove a version of Ihara's theorem.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524002763/pdfft?md5=cac16eb3054eb6231f0700f90e0e55f9&pid=1-s2.0-S0024379524002763-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141541163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On non-bipartite graphs with strong reciprocal eigenvalue property","authors":"Sasmita Barik ,&nbsp;Rajiv Mishra ,&nbsp;Sukanta Pati","doi":"10.1016/j.laa.2024.06.023","DOIUrl":"10.1016/j.laa.2024.06.023","url":null,"abstract":"<div><p>Let <em>G</em> be a simple connected graph and <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the adjacency matrix of <em>G</em>. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is nonsingular and <span><math><mi>X</mi><mo>=</mo><mi>S</mi><mi>A</mi><msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> is entrywise nonnegative for some signature matrix <em>S</em>, then <em>X</em> can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of <em>G</em>, denoted by <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>. A graph <em>G</em> is said to have the reciprocal eigenvalue property (property(R)) if <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is nonsingular, and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>λ</mi></mrow></mfrac></math></span> is an eigenvalue of <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> whenever <em>λ</em> is an eigenvalue of <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Further, if <em>λ</em> and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>λ</mi></mrow></mfrac></math></span> have the same multiplicity for each eigenvalue <em>λ</em>, then <em>G</em> is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree <em>T</em>, the following conditions are equivalent: a) <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> is isomorphic to <em>T</em>, b) <em>T</em> has property (R), c) <em>T</em> has property (SR) and d) <em>T</em> is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).</p><p>Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> exists and <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> is isomorphic to <em>G</em>. It follows that each graph <em>G</em> in this class has property (SR).</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513565","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":"Products of infinite upper triangular quadratic matrices","authors":"M.H. Bien ,&nbsp;V.M. Tam ,&nbsp;D.C.M. Tri ,&nbsp;L.Q. Truong","doi":"10.1016/j.laa.2024.06.021","DOIUrl":"10.1016/j.laa.2024.06.021","url":null,"abstract":"<div><p>Let <em>F</em> be a field and <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> a quadratic polynomial in <span><math><mi>F</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> with <span><math><mi>q</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>≠</mo><mn>0</mn></math></span>. We denote by <span><math><msub><mrow><mi>T</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> the algebra of all infinite upper triangular matrices over the field <em>F</em>. A matrix <span><math><mi>A</mi><mo>∈</mo><msub><mrow><mi>T</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is called a quadratic matrix with respect to <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> if <span><math><mi>q</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. In this paper, we first investigate the subgroup in <span><math><msub><mrow><mi>T</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> generated by all quadratic matrices with respect to <span><math><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and then present some applications.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices","authors":"Sébastien Bossu","doi":"10.1016/j.laa.2024.06.018","DOIUrl":"10.1016/j.laa.2024.06.018","url":null,"abstract":"<div><p>A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented, and the numerical stability of the formulas is studied.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510215","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":"Distribution of signless Laplacian eigenvalues and graph invariants","authors":"Leyou Xu,&nbsp;Bo Zhou","doi":"10.1016/j.laa.2024.06.019","DOIUrl":"https://doi.org/10.1016/j.laa.2024.06.019","url":null,"abstract":"<div><p>For a simple graph on <em>n</em> vertices, any of its signless Laplacian eigenvalues is in the interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mo>]</mo></math></span>. In this paper, we give relationships between the number of signless Laplacian eigenvalues in specific intervals in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn><mo>]</mo></math></span> and graph invariants including matching number and diameter.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481560","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":"Minimal affine varieties of superalgebras with superinvolution: A characterization","authors":"Onofrio M. Di Vincenzo ,&nbsp;Vincenzo C. Nardozza","doi":"10.1016/j.laa.2024.06.020","DOIUrl":"https://doi.org/10.1016/j.laa.2024.06.020","url":null,"abstract":"<div><p>We exhibit a class <span><math><mi>C</mi></math></span> of finite dimensional algebras with superinvolution over an algebraically closed field of characteristic zero, with the remarkable property that each member of <span><math><mi>C</mi></math></span> generates a minimal variety of algebras with superinvolution. This sums up to the fact that any affine minimal variety of algebras with superinvolution is generated by a suitable member of <span><math><mi>C</mi></math></span>, thus providing a complete characterization of the affine minimal varieties of algebras with superinvolution.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S002437952400274X/pdfft?md5=036a5bb31a29f608772f005e9a4e7256&pid=1-s2.0-S002437952400274X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141484914","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":"Spherical triangular configurations with invariant geometric mean","authors":"Luís Machado, Knut Hüper, Krzysztof Krakowski, Fátima Silva Leite","doi":"10.1016/j.laa.2024.06.017","DOIUrl":"https://doi.org/10.1016/j.laa.2024.06.017","url":null,"abstract":"The main objective is to characterize all configurations of three distinct points on the -dimensional sphere that have the same Riemannian geometric mean and find efficient ways to compute such invariant. The regular case, when the points form the vertices of an equilateral spherical triangle, appears as the global minimum of an appropriate cost function. As a warm-up, and also to get more insight for the spherical case, we first develop our ideas for configurations in the Euclidean space . In both cases, the theoretical results are supported by numerical experiments and illustrated by meaningful plots.","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513564","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":"Minimal graphs with eigenvalue multiplicity of n − d","authors":"Yuanshuai Zhang ,&nbsp;Dein Wong ,&nbsp;Wenhao Zhen","doi":"10.1016/j.laa.2024.06.015","DOIUrl":"https://doi.org/10.1016/j.laa.2024.06.015","url":null,"abstract":"<div><p>For a connected graph <em>G</em> with order <em>n</em>, let <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the number of its distinct eigenvalues and <em>d</em> be the diameter. We denote by <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> the eigenvalue multiplicity of <em>μ</em> in <em>G</em>. It is well known that <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>, which shows <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>μ</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span> for any real number <em>μ</em>. A graph is called <span><math><mi>m</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>m</mi><mi>a</mi><mi>l</mi></math></span> if <span><math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></math></span>. In 2013, Wong et al. characterize all minimal graphs with <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span>. In this paper, by applying the star complement theory, we prove that if <em>G</em> is not a path and <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>μ</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span>, then <span><math><mi>μ</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>}</mo></math></span>. Furthermore, we completely characterize all minimal graphs with <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mi>d</mi></math></span>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141480396","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":"Strassen's rank additivity for small tensors, including tensors of rank less or equal 7","authors":"Filip Rupniewski","doi":"10.1016/j.laa.2024.06.016","DOIUrl":"https://doi.org/10.1016/j.laa.2024.06.016","url":null,"abstract":"<div><p>The article is concerned with the problem of additivity of the tensor rank. That is, for two independent tensors, we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we present families of pairs of small three-way tensors for which the additivity holds. For instance, over the base field <span><math><mi>C</mi></math></span>, it is the case if both tensors are of rank less or equal 7. This proves that a pair of <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrix multiplication tensors has the rank additivity property. We show also that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524002702/pdfft?md5=1db3e05ac9a233539b0cf0c1e3838556&pid=1-s2.0-S0024379524002702-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141439251","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":"Corner replacement for Morita contexts","authors":"Raphael Bennett-Tennenhaus","doi":"10.1016/j.laa.2024.06.013","DOIUrl":"10.1016/j.laa.2024.06.013","url":null,"abstract":"<div><p>We consider how Morita equivalences are compatible with the notion of a corner subring. Namely, we outline a canonical way to replace a corner subring of a given ring with one which is Morita equivalent, and look at how such an equivalence ascends.</p><p>We use the language of Morita contexts, and then specify these more general results. We give applications to trivial extensions of finite-dimensional algebras, tensor rings of pro-species, semilinear clannish algebras arising from orbifolds, and functor categories.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524002660/pdfft?md5=90052d2c08523ea1fa95e982f067acb0&pid=1-s2.0-S0024379524002660-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141402534","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}