查看更多>>摘要：& nbsp;In this paper, the generalized centralizer group U-n(A) of a tropical n x n matrix A and the centralizer group P-n(E) of a tropical idempotent normal matrix E are introduced and studied. It is proved that U(n()A) is a product of two specific normal subgroups. And a structural description of P-n(E) is given when E is not strongly regular. It is also made some observations on E when P-n(E) is isomorphic to a 2-closed transitive permutation group on {1, 2, ... , n}. (C)& nbsp;2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. A generalized Bethe tree with k levels is a rooted tree in which vertices at the same level have the same degree. Fransa and Brondani (2021) [4] recently conjectured that any generalized Bethe tree with k levels has exactly k main eigenvalues whenever k is even. We disprove the conjecture by constructing a family of counterexamples for even integers k >= 6.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：This article reviews and discusses a geometric perspective on the well-known fact in graph theory that the effective resistance is a metric on the nodes of a graph. The classical proofs of this fact make use of ideas from electrical circuits or random walks; here we describe an alternative approach which combines geometric (using simplices) and algebraic (using the Schur complement) ideas. These perspectives are unified in a matrix identity of Miroslav Fiedler, which beautifully summarizes a number of related ideas at the intersection of graphs, Laplacian matrices and simplices, with the metric property of the effective resistance as a prominent consequence.(c) 2022 Elsevier Inc. All rights reserved.
Greaves, Gary R. W.Iverson, Joseph W.Jasper, JohnMixon, Dustin G....
31页
查看更多>>摘要：We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but 18 known SRG parameters on v <= 1300 vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension 5.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：We revisit the block-Jacobi iterative solver and preconditioner for Hermitian positive definite systems of equations. In order to force convergence of the iterative scheme, the diagonal blocks are nested hierarchically in a binary fashion. The resulting parallel technique, the hierarchical binary Jacobi (hbJ), fits within the nested matrix splitting framework. The convergence criteria of the hbJ are provided and its computational complexity is studied. Comparison theorems based on the levels of the nested iteration are given. A computational study for its use as iterative solver and preconditioner is performed within the class of Wishart random generated Hermitian positive definite matrices. Counterexamples are provided for a selection of desired comparison results.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：A B S T R A C T Perfect state transfer (PST) has great importance due to its applications in quantum information processing, quantum communication networks and cryptography. In the present work, we establish a characterization of Cayley graphs over dicyclic groups T-4n, having PST. (C)& nbsp;2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The bottleneck matrix M of a rooted tree T is a combinatorial object encoding the spatial distribution of the vertices with respect to the root. The spectral radius of M, known as the Perron value of the rooted tree, is closely related to the theory of the algebraic connectivity. In this paper, we investigate the Perron values of various classes of rooted trees by making use of combinatorial and linear-algebraic techniques. This results in multiple bounds on the Perron values of these classes, which can be straightforwardly applied to provide information on the algebraic connectivity. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
查看更多>>摘要：An element g in a group G is called reversible (or real) if it is conjugate to g(-1) in G, i.e., there exists h in G such that g(-1) = hgh-1. The element g is called strongly reversible if the conjugating element h is an involution (i.e., element of order at most two) in G. In this paper, we classify reversible and strongly reversible elements in the isometry groups of F-Hermitian spaces, where F = C or H. More precisely, we classify reversible and strongly reversible elements in the groups Sp(n) alpha H-n, U(n) alpha C-n and SU(n) alpha C-n. We also give a new proof of the classification of strongly reversible elements in Sp(n). (c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper, we propose and solve low phase-rank approximation problems, which serve as a counterpart to the wellknown low-rank approximation problem and the SchmidtMirsky theorem. It is well known that a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix being defined as the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with the matrix arithmetic mean, it turns out that a natural parallel for the low phase-rank approximation problem is to use the matrix geometric mean to measure the approximation error. Importantly, we derive a majorization inequality between the phases of the geometric mean and the arithmetic mean of the phases, similarly to the Ky-Fan inequality for eigenvalues of Hermitian matrices. A characterization of the solutions to the proposed problem, with the same flavor as the Schmidt-Mirsky theorem, is then obtained in the case where both the objective matrix and the approximant are restricted to be positive-imaginary. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary.(c) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin (2016) [29] and improves a result of Kuai (2018) [17]. Moreover, we provide a unified treatment of a result of Ando (2014) [2] and a recent result of Li et al. (2021) [22]. Furthermore, we also extend some determinant inequalities involving partial traces to a larger class of matrices whose numerical ranges are contained in a sector. In addition, some extensions on trace inequalities for positive semidefinite 2 x 2 block matrices are also included.(c) 2022 Elsevier Inc. All rights reserved.
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