Abstract We characterize the idempotent stable range 1, 2 × 2 matrices over commutative rings and in particular the integral matrices with this property. Several special cases and examples complete the subject.
{"title":"On idempotent stable range 1 matrices","authors":"G. Călugăreanu, Horia F. Pop","doi":"10.1515/spma-2022-0159","DOIUrl":"https://doi.org/10.1515/spma-2022-0159","url":null,"abstract":"Abstract We characterize the idempotent stable range 1, 2 × 2 matrices over commutative rings and in particular the integral matrices with this property. Several special cases and examples complete the subject.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"251 - 257"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46096618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let Λ = {λ1, λ2, . . ., λn} be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λi ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.
{"title":"Spectra inhabiting the left half-plane that are universally realizable","authors":"R. Soto","doi":"10.1515/spma-2021-0155","DOIUrl":"https://doi.org/10.1515/spma-2021-0155","url":null,"abstract":"Abstract Let Λ = {λ1, λ2, . . ., λn} be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λi ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"180 - 192"},"PeriodicalIF":0.5,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46866808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Bolla, Vilas Winstein, Tao You, Frank Seidl, Fatma Abdelkhalek
Abstract We discuss spectral clustering from a variety of perspectives that include extending techniques to rectangular arrays, considering the problem of discrepancy minimization, and applying the methods to directed graphs. Near-optimal clusters can be obtained by singular value decomposition together with the weighted kk-means algorithm. In the case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, while in the case of edge-weighted graphs, a normalized Laplacian-based clustering. In the latter case, it is proved that a spectral gap between the (k−1)left(k-1)st and kkth smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2k−1{2}^{k-1}, but only the first k−1k-1 eigenvectors are used in the representation. The ensemble of these eigenvectors constitute the so-called Fiedler-carpet.
{"title":"Regularity-based spectral clustering and mapping the Fiedler-carpet","authors":"M. Bolla, Vilas Winstein, Tao You, Frank Seidl, Fatma Abdelkhalek","doi":"10.1515/spma-2022-0167","DOIUrl":"https://doi.org/10.1515/spma-2022-0167","url":null,"abstract":"Abstract We discuss spectral clustering from a variety of perspectives that include extending techniques to rectangular arrays, considering the problem of discrepancy minimization, and applying the methods to directed graphs. Near-optimal clusters can be obtained by singular value decomposition together with the weighted kk-means algorithm. In the case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, while in the case of edge-weighted graphs, a normalized Laplacian-based clustering. In the latter case, it is proved that a spectral gap between the (k−1)left(k-1)st and kkth smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2k−1{2}^{k-1}, but only the first k−1k-1 eigenvectors are used in the representation. The ensemble of these eigenvectors constitute the so-called Fiedler-carpet.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"394 - 416"},"PeriodicalIF":0.5,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45718718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.
{"title":"The dual of number sequences, Riordan polynomials, and Sheffer polynomials","authors":"T. He, J. L. Ramírez","doi":"10.1515/spma-2021-0153","DOIUrl":"https://doi.org/10.1515/spma-2021-0153","url":null,"abstract":"Abstract In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"153 - 165"},"PeriodicalIF":0.5,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48951428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai,i+1ai+1,i= 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.
{"title":"On some reciprocal matrices with elliptical components of their Kippenhahn curves","authors":"Muyan Jiang, I. Spitkovsky","doi":"10.1515/spma-2021-0151","DOIUrl":"https://doi.org/10.1515/spma-2021-0151","url":null,"abstract":"Abstract By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai,i+1ai+1,i= 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"117 - 130"},"PeriodicalIF":0.5,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41857790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.
{"title":"Graphs with the second signless Laplacian eigenvalue ≤ 4","authors":"S. Drury","doi":"10.1515/spma-2021-0152","DOIUrl":"https://doi.org/10.1515/spma-2021-0152","url":null,"abstract":"Abstract We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"131 - 152"},"PeriodicalIF":0.5,"publicationDate":"2021-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43551306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We define GG-cospectrality of two GG-gain graphs (Γ,ψ)left(Gamma ,psi ) and (Γ′,ψ′)left(Gamma ^{prime} ,psi ^{prime} ), proving that it is a switching isomorphism invariant. When GG is a finite group, we prove that GG-cospectrality is equivalent to cospectrality with respect to all unitary representations of GG. Moreover, we show that two connected gain graphs are switching equivalent if and only if the gains of their closed walks centered at an arbitrary vertex vv can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph ΓGamma with nn vertices and mm edges, is equal to the number of simultaneous conjugacy classes of the group Gm−n+1{G}^{m-n+1}. We provide examples of GG-cospectral switching nonisomorphic graphs and we prove that any gain graph on a cycle is determined by its GG-spectrum. Moreover, we show that when GG is a finite cyclic group, the cospectrality with respect to a faithful irreducible representation implies the cospectrality with respect to any other faithful irreducible representation, and that the same assertion is false in general.
{"title":"On cospectrality of gain graphs","authors":"Matteo Cavaleri, A. Donno","doi":"10.1515/spma-2022-0169","DOIUrl":"https://doi.org/10.1515/spma-2022-0169","url":null,"abstract":"Abstract We define GG-cospectrality of two GG-gain graphs (Γ,ψ)left(Gamma ,psi ) and (Γ′,ψ′)left(Gamma ^{prime} ,psi ^{prime} ), proving that it is a switching isomorphism invariant. When GG is a finite group, we prove that GG-cospectrality is equivalent to cospectrality with respect to all unitary representations of GG. Moreover, we show that two connected gain graphs are switching equivalent if and only if the gains of their closed walks centered at an arbitrary vertex vv can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph ΓGamma with nn vertices and mm edges, is equal to the number of simultaneous conjugacy classes of the group Gm−n+1{G}^{m-n+1}. We provide examples of GG-cospectral switching nonisomorphic graphs and we prove that any gain graph on a cycle is determined by its GG-spectrum. Moreover, we show that when GG is a finite cyclic group, the cospectrality with respect to a faithful irreducible representation implies the cospectrality with respect to any other faithful irreducible representation, and that the same assertion is false in general.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"343 - 365"},"PeriodicalIF":0.5,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42313905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The spread of a graph is the difference between the largest and most negative eigenvalue of its adjacency matrix. We show that for sufficiently large nn, the nn-vertex outerplanar graph with maximum spread is a vertex joined to a linear forest with Ω(n)Omega left(n) edges. We conjecture that the extremal graph is a vertex joined to a path on n−1n-1 vertices.
{"title":"On the spread of outerplanar graphs","authors":"D. Gotshall, M. O’Brien, Michael Tait","doi":"10.1515/spma-2022-0164","DOIUrl":"https://doi.org/10.1515/spma-2022-0164","url":null,"abstract":"Abstract The spread of a graph is the difference between the largest and most negative eigenvalue of its adjacency matrix. We show that for sufficiently large nn, the nn-vertex outerplanar graph with maximum spread is a vertex joined to a linear forest with Ω(n)Omega left(n) edges. We conjecture that the extremal graph is a vertex joined to a path on n−1n-1 vertices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"299 - 307"},"PeriodicalIF":0.5,"publicationDate":"2021-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42274993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Explicit expressions for the coefficients of the group inverse of a circulant matrix depending on four complex parameters are analytically derived. The computation of the entries of the group inverse are now reduced to the evaluation of a polynomial. Moreover, our methodology applies to both the invertible and the singular case, the latter being computationally less expensive. The techniques we use are related to the solution of boundary value problems associated with second order linear difference equations.
{"title":"The group inverse of circulant matrices depending on four parameters","authors":"Á. Carmona, A. Encinas, M. Jiménez, M. Mitjana","doi":"10.1515/spma-2021-0149","DOIUrl":"https://doi.org/10.1515/spma-2021-0149","url":null,"abstract":"Abstract Explicit expressions for the coefficients of the group inverse of a circulant matrix depending on four complex parameters are analytically derived. The computation of the entries of the group inverse are now reduced to the evaluation of a polynomial. Moreover, our methodology applies to both the invertible and the singular case, the latter being computationally less expensive. The techniques we use are related to the solution of boundary value problems associated with second order linear difference equations.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"87 - 108"},"PeriodicalIF":0.5,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42980803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let R be a commutative finite local ring. Two enumeration problems over R are presented. We enumerate the matrices over R with a given McCoy rank and a given number of rows of single unit, and the free linear codes over R which have a given rank and a given number of vectors of single unit.
{"title":"Enumeration of some matrices and free linear codes over commutative finite local rings","authors":"S. Sirisuk","doi":"10.1515/spma-2021-0150","DOIUrl":"https://doi.org/10.1515/spma-2021-0150","url":null,"abstract":"Abstract Let R be a commutative finite local ring. Two enumeration problems over R are presented. We enumerate the matrices over R with a given McCoy rank and a given number of rows of single unit, and the free linear codes over R which have a given rank and a given number of vectors of single unit.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"109 - 116"},"PeriodicalIF":0.5,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45724429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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