The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of some associated graphs. We introduce commutatively closed graphs and investigate properties of commutatively closed subsets of a ring with the help of graph theory. In particular, we compute the diameter of matrix rings and Artinian semisimple algebras.

Let H be an infinite dimensional Hilbert space and B(H) be the algebra of all bounded linear operators on H. For A∈B(H), let σ∗(A) denote the semi-Fredholm domain, the Fredholm domain or the Weyl domain of A in the spectrum, σ(A). We characterize the form of a map ϕ from B(H) into itself with range contains the operators of rank at most two and satisfies σ∗(ϕ(A)ϕ(B)∗)=σ∗(AB∗) for all A,B∈B(H). We also

In this work, we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and

For polyhedral cones in the Euclidean space, we present its conic dimension, which is invariant under linear isomorphisms that is sensitive to the number of generators of this cone, and the related notion of conic basis. We may interpret these two notions as versions of the definitions of linear dimension and linear basis for linear subspaces in the setting of polyhedral cones. We establish a conic

We study the representations of two special Jordan triple systems (with respect to the product xyz + zyx): The first is the Jordan triple system JS of all symmetric n by n(n≥2) matrices over a field F of characteristic zero, the second is the Jordan triple system JH of all Hermitian n by n(n≥2) matrices over the complex numbers C. We show that the universal (associative) envelope of JS is isomorphic

In 2020, Ferreyra et al. defined the weak core inverse of a complex matrix. We generalize it to a unitary ring with involution and investigate the relationship between the weak core inverse and the m-weak group inverse. Furthermore, we prove that an element is pseudo core invertible if this element is both {1,3}-invertible and weak group invertible. Then, we show that each nilpotent element is Moore–Penrose

Let V be a vector space with countable dimension over a field, and let u be an endomorphism of it which is locally finite, i.e.  x,u(x),u2(x),… are linearly dependent for all x in V. We give several necessary and sufficient conditions for the decomposability of u into the sum of two square-zero endomorphisms. Moreover, if u is invertible, we give necessary and sufficient conditions for the decomposability

Euclidean distance matrices ( EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices ( GEDMs) that include EDMs. Each GEDM is an entry-wise nonnegative matrix. A GEDM is not symmetric unless it is an EDM. By some new techniques, we show that many significant results on Euclidean

This paper is concerned with the lower and upper redundancy of infinite frames in a separable Hilbert space. Using unit norm linear operators, a new representation of upper redundancy is given. Also, we show that there is no frame such that its lower redundancy is less than one and its upper redundancy is one. Moreover, for given (μ,λ)∈M:={(μ,λ);1≤⌈μ⌉≤⌊λ⌋}∖{(μ,1);0<μ<1}, we construct a frame such that

ABSTRACT Let H,〈⋅,⋅〉 be a complex Hilbert space and A be a positive bounded linear operator on H. The semi-inner product 〈x,y〉A:=〈Ax,y〉,x,y∈H, induces a semi-norm ∥⋅∥A on H. Let ωA(T) and ∥T∥A denote the A-numerical radius and the A-operator semi-norm of an operator T in semi-Hilbertian space H,〈⋅,⋅〉A, respectively. In this paper, some new bounds for the A-numerical radius of operators in semi-Hilbertian

Let G be a simple connected graph and A(G) be the adjacency matrix of G. Then G is said to have the anti-reciprocal eigenvalue property (property (−R)) if −1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and −1λ have the same multiplicity for each eigenvalue λ, then it is said to have the strong anti-reciprocal eigenvalue property (property (−SR)). A graph G is called

Let G be a simple graph of order n. The matrix S(G)=D(G)+A(G) is called the signless Laplacian matrix of G, where D(G) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let s1(G) and λ1(G) be the largest eigenvalue of S(G) and A(G), respectively. In this paper, we first present sharp upper and lower bounds for s1(G) and λ1(G) involving the maximum degree

ABSTRACT We obtain two matrix inequalities on the off-diagonal block of 2×2 block positive semidefinite matrices and the geometric mean of its diagonal blocks. They improve some results of Lee [Linear Algebra Appl. 2015:486;449–453] and extend some inequalities of Fujimoto and Seo [Linear Multilinear Algebra. 2018:66;1564–1577]. Applications to sector matrices are given.

An irreducible module for the parafermion vertex operator algebra K(sl2,k) is said to be of σ-type if an automorphism of the fusion algebra of K(sl2,k) of order k is trivial on it. For any integer k≥3, we show that there exists an automorphism of order 2 of the subalgebra of the fusion algebra of K(sl2,k)〈θ〉 spanned by the irreducible direct summands of σ-type irreducible K(sl2,k)-modules, where θ

We study the relationship between ergodicity and irreducibility of transition probability tensors. We then introduce the notions of first passage times and mean first passage times, both in the tensor form, for higher order Markov chains and prove some properties of these quantities. Numerical examples are also given to illustrate our results.

It was shown in 2006 that among the nonsingular trees T (whose adjacency matrix A(T) is nonsingular), the corona trees (trees that are obtained by taking any tree T and then adding a new pendant vertex at each vertex of T) are the only ones which satisfy the reciprocal eigenvalue property (λ is an eigenvalue of A(T) if and only if 1λ is an eigenvalue of A(T), where their multiplicities are allowed

Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e. it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect the separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem

We introduce an equivalence relation ρ on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving ρ. Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves ρ. Finally

Complex conference matrices have received considerable attention in the last few years due to their application in quantum information theory and in geometry. Various results and constructions are known for symmetric and Hermitian conference matrices. In this article, we deal mainly with constructions of complex skew-symmetric conference matrices. We classify all complex conference matrices up to order

In this note, we establish the cartesian decomposition of C-normal operator on a complex separable Hilbert space. Using this we give two characterizations of C-normal operators. These characterizations are useful in checking C-normality of an operator. We illustrate these results with examples.

The Hermitian and skew-Hermitian splitting (HSS) method is a well-known method employed in solving the non-Hermitian positive definite linear systems. However, the performance of the HSS method highly depends on the parameter value. In this paper, we address this issue by introducing two practical selections of this parameter, which are computed based on the Rayleigh quotient of the Hermitian part

We provide a novel recursive method, which does not require any assumption, to compute the entries of the kth power of a semicirculant matrix by using a closed-form expression for the mth term of a certain recursive polynomial sequence. As an application, a method for computing the entries of the kth power of r-circulant matrices is also presented.

In this article, we introduce a q-analogue of the operators defined by Usta and Betus (Linear Multilinear Algebra, DOI: 10.1080/03081087.2020.1791033 (2020)) and find a general expression of moments for the q- operators. Next, we study the approximation properties of these operators in a weighted space via A-statistical method and the power series method. Later, we define a k-th order generalization

This work mainly explores the idea of the spectrum of the fractional difference operator Δν(α) (of order α∈R) over the Banach space ℓ1. The difference operator Δν(α) via the sequence x=(xk) is defined by (Δν(α)x)k=∑i=0∞(−α)ii!νk−ixk−i,where (α)i denotes the Pochhammer symbol, α∈R and ν=(νk) is either a constant or strictly decreasing sequence of positive real numbers with some conditions. Sets comprising

This article deals with necessary and sufficient conditions for a family of elements in a Euclidean Jordan algebra to have simultaneous (order) spectral decomposition. Motivated by a well-known matrix theory result that any family of pairwise commuting complex Hermitian matrices is simultaneously (unitarily) diagonalizable, we show that in the setting of a general Euclidean Jordan algebra, any family

ABSTRACT Let A be a C∗-algebra with unit I and let S(A) be the state space of A. Let A∈A, an element f∈S(A) is said to be maximal for A if f(A∗A)=‖A‖2. Denote the set of all maximal states for A by Smax(A) and define the algebraic maximal numerical range of A as follows V0(A):={f(A):f∈Smax(A)}. We say that A is orthogonal to I in the Birkhoff–James sense, written as A⊥BJI, whenever ‖A‖≤‖A−λ‖, for all

As one application of the Cauchy–Binet formula, the Ball–Barthe inequality plays a key role in solving reverse isoperimetric inequalities. Recently, a new Grassmannian Ball–Barthe inequality was established. With it, reverse isoperimetric inequalities on Grassmann manifolds were successfully proved. In this paper, we will establish its matrix form with a simpler proof than that of Grassmannian form

In this paper, the super-biderivations on the super Heisenberg-Virasoro algebra are completely determined. We find that there exist non-inner super-skewsymmetric super-biderivations of the super Heisenberg-Virasoro algebra.

Quantum coherence has recently emerged as a key candidate for use as a resource in various quantum information processing tasks. One of the main problems in any quantum resource theory is the characterization of the conversions between resources by means of the free operations of the theory. In this work, we ask if coherence, already present in a reference system, can be broadcast to the auxiliary

Schur's inequality for the sum of products of the differences of real numbers states that for x,y,z,t≥0, xt(x−y)(x−z)+yt(y−z)(y−x)+zt(z−x)(z−y)≥0. In this paper, we study a generalization of this inequality to more terms, more general functions of the variables and algebraic structures such as vectors and Hermitian matrices.

ABSTRACT Identifiability holds for the k-secant variety σk(X) of an embedded variety X⊂Pr if a general q∈σk(X) is in the linear span of a unique subset of X with cardinality k. Identifiability is true if the general tangential k-contact locus Γk⊂X has dimension 0. Under certain assumptions on dim⁡σx(X) for some specific x>k, we get dim⁡Γk=0. Here we give some conditions which exclude the case Γx a

In this paper, we study 2-local superderivations on the Lie superalgebra of Block type S(q), which is an infinite dimensional Lie superalgebra with an outer derivation. We prove that all 2-local superderivations on S(q) are superderivations.

Let L be a finite-dimensional nilpotent Lie algebra of class two over a field F of characteristic different from two. The aim of the present paper is to give an upper bound for the dimension of the center factor L/Z(L) of a capable Lie algebra L. As a result, we also find a lower bound for the dimension of the derived subalgebra L2 for a capable Lie algebra L. Furthermore, we also give some criteria

In this paper, we propose a feasible and effective iteration method for solving the tensor equation A∗NX∗MB=C with a tensor inequality constraint D∗NX∗ME≥F. The global convergence is established. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed algorithm.

In this paper, we first investigate some properties of an orthogonal complement of a subsemimodule and give some necessary and sufficient conditions for every subsemimodule to be equal to the orthogonal complement of its orthogonal complement. Then we obtain some necessary and sufficient conditions for a set of standard orthogonal vectors to be extended to a standard orthogonal basis of a semimodule

Let R be a commutative ring, m>1 an integer and q>1 an odd number. The following questions are addressed: (1) when is every element in Mn(R) and, respectively, in Tn(R) a product of q-potents? (2) when is every element in Mn(R) and, respectively, in Tn(R) a product of m q-potents? and (3) when is every element in Mn(R) and, respectively, in Tn(R) a product of an idempotent and a q-potent?

ABSTRACT The formula for the spectral moment of graphs which is represented by the number of subgraphs is called the RNS spectral moment formula of graphs. In this paper, the research on the RNS spectral moment formula of graphs is generalized to hypergraphs. We give a formula for the spectral moment of a hypertree in terms of the number of some subhypertrees. Using this formula, we obtain the first

Perfect quantum state transfer plays a crucial role in quantum information processing and quantum computation. There has been extensive study of perfect quantum state transfer on Cayley graphs over abelian groups. In this paper, we consider the existence of perfect quantum state transfer on Cayley graphs over semi-dihedral groups which are non-abelian groups. Using the representations of semi-dihedral

ABSTRACT The symmetric Pascal matrix is a square matrix whose entries are given by binomial coefficients modulo 2. In 1997, Christopher and Kennedy defined and studied the binomial graph, which is the graph whose adjacency matrix is the symmetric Pascal matrix. They computed the spectrum of the binomial graph of order a power of 2. In this paper, we study spectral properties of the binomial graph of

Let P and Q be projections on a Hilbert C∗-module. This paper deals with the orthogonal complementarity of certain range closures associated with P and Q. Some necessary and sufficient conditions are derived under which (P,Q) is harmonious.

The notions of C-tensor, C0-tensor and C¯-tensor are introduced first. Different necessary and sufficient conditions for a tensor to be a C-tensor, C0-tensor and C¯-tensor are provided. We next show that the sum of two C-tensors ( C0-tensors) is a C-tensor ( C0-tensor) while the Hadamard product of two C-tensors ( C0-tensors) is not a C-tensor ( C0-tensor). We also present a result that illustrates

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in Y∗ and when this association forms a linear operator of norm-1 from Y∗ to X∗. It is proved that, under certain geometric assumptions on X,Y,Z, these properties are stable with respect to the injective

In this paper, we present results concerning orthogonality in Hilbert C∗-modules. Moreover, for a C∗-algebra A, we prove theorems concerning the multi- A-linearity and its preservation by A-linear operators. New version of solution of the orthogonality equation on Hilbert C∗-modules and mappings preserving orthogonality are also investigated.

In this paper, a novel method is presented to compute an explicit formula for the inverse of the confluent Vandermonde matrices. The proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences, notably on the situations where the Vandermonde matrices have acquired much usefulness. The method presented here is direct and straightforward, it gives explicit

The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by Gn,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984–996] conjectured

In this paper, first we give the cohomology theory of a 3-Lie algebra with a representation (called a 3- LieRep pair) and study the linear deformation theory of a 3- LieRep pair. We introduce the notion of a Nijenhuis structure on a 3- LieRep pair, which generates a trivial linear deformation. Then by adding compatibility conditions between Nijenhuis structures and relative Rota-Baxter operators, we

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight ±i and with a unique perfect matching, denoted by Ug, is considered in this article. We characterize the 3-coloured digraphs in Ug whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs

Let G be a connected hypergraph. The distance spectral radius of G is the largest eigenvalue of its distance matrix. The Wiener index of G is defined to be the sum of distances between every unordered pair of vertices of G. A connected k-uniform hypergraph G with n vertices and m edges is called bicyclic if n=m(k−1)−1. Firstly, we obtain a lower bound on the Wiener index of k-uniform bicyclic hypergraphs

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can

This article aims to interpret some Lie triple derivations of Tensor algebras T⊗F(a,bF), using generalized quaternion algebra (a,bF) over a field F and assuming T as an F-linear associative algebra. The study concludes with complete description of all Lie triple derivations of A⊗FL. It is pertinent to mention here that A is assumed to be a finite-dimensional commutative algebra and L is a simple Lie

S.-Z. Song et. al. [Linear and Multilinear Algebra. 2016;64(7):1258–1265] studied linear operators from Sm to Sn that preserve cp-rank, where Sn denotes the space of n×n real symmetric matrices. One of the results states that a linear operator ϕ:Sm→Sn preserves cp-rank if and only if ϕ(X)=STXS for some nonnegative matrix S with a full row rank. We show here that while this statement is indeed valid

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct

Let Kn be the second-order cone in Rn, where n≥3. Given a vector q∈Rn and an n×n matrix G, the second order cone linear complementarity problem SOLCP(G,q) is to find a vector x∈Rn such that x∈Kn,y:=Gx+q∈KnandxTy=0.The matrix G is said to have the  Q-property if SOLCP(G,q) has a solution for all q∈Rn. An n×n matrix G is called a cone automorphism if GKn=Kn. In this paper, we obtain a simple characterization

A conjecture by Erds and Hajnal [Ramsey-type theorems. Discrete Appl Math. 1989;25:37–52] pointed out that the structure of graphs with forbidden induced subgraphs is very different from general graphs. More precisely, they contain much larger cliques or independent sets. Inspired by this, we prove that non-Hamilton-connected graphs satisfying certain conditions contain a large clique. From the perspective

Among all simple 2-connected graphs, and among all θ-graphs, the graphs with the minimum algebraic connectivity are completely determined, respectively.

In this paper, we are concerned with a class of tensor equations. Some explicit solutions to these equations are obtained. They are extensions of some results given by Lancaster. At the end of the article, these results are also employed to prove Lyapunov and Stein stability theorems for tensors.

In this paper, we prove the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1)(0

A graph is supereulerian if it contains a spanning closed trail. We prove several Erds-type extremal size conditions with a lower bounded minimum degree for a graph to be supereulerian and with different edge-connectivity, with the corresponding extremal graphs characterized. These results are then applied to prove sufficient conditions involving adjacency spectral radius and signless Laplacian spectral

Several trace inequalities for positive semi-definite 2×2 block matrices are revisited. We present shorter proofs of these inequalities along with their refinements. We obtain bounds for the smallest and the largest eigenvalues of a positive semi-definite 2×2 block matrix involving the traces of its blocks.

In this paper, we continue the study of Berkani's property (UWΠ) introduced in [Berkani M, Kachad M. New Browder and Weyl type theorems. Bull Korean Math Soc. 2015;52:439–452.], in connection with other Weyl-type theorems. Moreover, we give counterexamples to show that some recent results related to this property, which are announced and proved by P. Aiena and M. Kachad in [Aiena P, Kachad M. Property