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Abstract: In this paper we introduce a class of regular bipartite graphs whose biadjacency matrices are circulant matrices – a generalization of circulant graphs which happen to be bipartite – and we describe some of their properties. Notably, we compute upper and lower bounds for the zero forcing number for such a graph based only on the parameters that describe its biadjacency matrix. The main results of the paper characterize the bipartite circulant graphs that achieve equality in the lower bound and compute their minimum ranks. [Copyright &y& Elsevier]

Abstract In this paper the spherical quasi-convexity of quadratic functions on spherically convex sets is studied. Several conditions characterizing the spherical quasi-convexity of quadratic functions are presented. In particular, conditions implying spherical quasi-convexity of quadratic functions on the spherical positive orthant are given. Some examples are provided as applications of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: Recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper. Some classical classes of matrices with quasiseparable structure include tridiagonal (related to real orthogonal polynomials) and banded matrices, unitary Hessenberg matrices (related to Szegö polynomials), and semiseparable matrices, as well as others. Hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results. Previous work has focused on characterizing -quasiseparable matrices, matrices with order-one quasiseparable structure that are also upper Hessenberg. In this paper, the authors introduce the concept of a twist transformation, and use such transformations to explain the relationship between -quasiseparable matrices and the subclass of -quasiseparable matrices (without the upper Hessenberg restriction) which are related to the same systems of polynomials. These results generalize the discoveries of Cantero, Fiedler, Kimura, Moral and Velázquez of five-diagonal matrices related to Horner and Szegö polynomials in the context of quasiseparable matrices. [Copyright &y& Elsevier]

In this paper we compute the critical group of the Kneser graph K G ( n , 2 ) . This is equivalent to computing the Smith normal form of a Laplacian matrix of this graph. [ABSTRACT FROM AUTHOR]

Let H n be the set of all n × n Hermitian matrices and H n m be the set of all m -tuples of n × n Hermitian matrices. For A = ( A 1 , . . . , A m ) ∈ H n m and for any linear map L : H n m → R ℓ , we define the L -numerical range of A by W L ( A ) : = { L ( U ⁎ A 1 U , . . . , U ⁎ A m U ) : U ∈ C n × n , U ⁎ U = I n } . In this paper, we prove that if ℓ ≤ 3 , n ≥ ℓ and A 1 , . . . , A m are simultaneously unitarily diagonalizable, then W L ( A ) is star-shaped with star center at L ( tr A 1 n I n , . . . , tr A m n I n ) . [ABSTRACT FROM AUTHOR]

Denote by μ k ( T ) the k th Laplacian eigenvalue of tree T . Let T n be the set of all trees of order n with perfect matchings. In this paper, we characterize the tree T in T 4 t with μ 2 ( T ) = t + 2 + t 2 + 4 2 , and the tree in T 6 t with μ 3 ( T ) = t + 2 + t 2 + 4 2 . These results affirm Problem of J.X. Li, W.C. Shiu, A. Chang (2010) [5] for the cases k = 2 and k = 3 , respectively. [ABSTRACT FROM AUTHOR]

In this paper we characterize the sets of all B ∈ B ( H ) such that AρB and the sets of all B ∈ B ( H ) such that BρA , where A ∈ B ( H ) is given and ρ ∈ { ≤ − , ≤ ⁎ , ≤ # , ≤ ⊕ } . [ABSTRACT FROM AUTHOR]

In this paper, we show that the primitive degree set of nonnegative primitive tensors with order m ( ≥ 3 ) and dimension n is { 1 , 2 , … , ( n − 1 ) 2 + 1 } , which implies that the results of the case m ≥ 3 (the case of tensors) is totally different from the case m = 2 (the case of matrices), and we also propose some open problems for further research. [ABSTRACT FROM AUTHOR]

In this paper we determine the unique graph with minimal least eigenvalue (of the adjacency matrix) within the set of connected graphs of fixed order n and size m , whenever m = ⌈ n 2 ⌉ ⌊ n 2 ⌋ + a , where a is a fixed integral constant in [ 1 , ⌈ n 2 ⌉ − 1 ] . [ABSTRACT FROM AUTHOR]

For a graph G = ( V , E ) with V = { 1 , 2 , … , n } , let S ( G ) be the set of all symmetric real n × n matrices A = [ a i , j ] with a i , j ≠ 0 , i ≠ j if and only if i j ∈ E . The inertia set of a graph G is the set of all possible inertias of matrices in S ( G ) . In this paper we give a formula that expresses the inertia set of a graph with a 2-separation in terms of the inertia sets of subgraphs of G . This formula follows from an extension to arbitrary fields with characteristic not equal to two. [ABSTRACT FROM AUTHOR]

A sign pattern (matrix) A is a matrix whose entries are from the set {+,-,0} . If no entry of A is zero, then A is called a full sign pattern. The inertia of a real matrix A is the ordered triple (n+,n-,n0), in which n+, n- and n0 are the numbers of its eigenvalues (counting multiplicities) with positive, negative and zero real parts, respectively. The refined inertia of A is the ordered 4-tuple (n+,n-,nz, 2np) where nz (resp., 2np) is the number of zero (resp., nonzero pure imaginary) eigenvalues of A. An nxn sign pattern A is an inertially arbitrary pattern (IAP) if it allows all possible inertias. Similarly, A is a refined inertially arbitrary pattern (rIAP) if it allows all possible refined inertias. A proper subset S of the set of all possible inertias (resp., refined inertia) of real matrices of order n is called a critical set of inertias (resp., critical set of refined inertias) for a family F of sign patterns of order n if for every A∈F, S⊆i(A) ensures that A is inertially arbitrary (resp., S⊆ri(A) ensures that A is refined inertially arbitrary); S is called a minimal critical set of inertias (resp., minimal critical set of refined inertias) for F if no proper subset of S is a critical set of inertias (resp., refined inertias) for F. In this paper, all minimal critical sets of inertias and refined inertias for 3x3 full sign patterns are identified. [ABSTRACT FROM AUTHOR]

Abstract: In Suzuki (1998) [7] Suzuki gave a classification of association schemes with multiple Q-polynomial structures, allowing for one exceptional case which has five classes. In this paper, we rule out the existence of this case. Hence Suzukiʼs theorem mirrors exactly the well-known counterpart for association schemes with multiple P-polynomial structures, a result due to Eiichi Bannai and Etsuko Bannai in 1980. [Copyright &y& Elsevier]

Abstract: Let be the set of all real matrices. An real matrix A is called circulant doubly stochastic if it is a convex combination of circulant permutation matrices. For , x is said to be circulant majorized by y (written as ), if there exists a circulant doubly stochastic matrix D such that . In this paper, the concept of circulant majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on . [Copyright &y& Elsevier]

Abstract: Let be a set of nonnegative matrices having at least one strictly positive product (all products with no ordering and with repetitions permitted). What is the minimal possible length of their positive product? In other words, what is the minimal number for which there are indices such that the matrix has all entries positive? In this paper we show, under some mild assumptions on matrices, that . We apply this result to estimate the rate of convergence in some limit theorems for random matrices. [Copyright &y& Elsevier]

Abstract: A signed graph is a pair , where is a graph (in which parallel edges are permitted, but loops are not) with and . The edges in are called odd edges and the other edges of E even. By we denote the set of all symmetric matrices with if i and j are adjacent and all edges between i and j are even, if i and j are adjacent and all edges between i and j are odd, if i and j are connected by even and odd edges, if and i and j are non-adjacent, and for all vertices i. The stable inertia set of a signed graph is the set of all pairs for which there exists a matrix with p positive and q negative eigenvalues which has the Strong Arnold Property. In this paper, we study the stable inertia set of (signed) graphs. [Copyright &y& Elsevier]

Abstract: This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the “approximate”, or “almost” zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the “approximate decoupling polynomials”. The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of “approximate decoupling polynomials”. [Copyright &y& Elsevier]

Abstract: By counting 1’s in the “right half” of consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth w. Then the matrix can be correctly centered and factored into block-diagonal permutation matrices. Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is determined “at infinity” in general, but from only rows for banded permutations. [Copyright &y& Elsevier]

Abstract: Cluster algebras of finite type is a fundamental class of algebras whose classification is identical to the famous Cartan–Killing classification. More recently, Fomin and Zelevinsky introduced another central notion of cluster algebras with principal coefficients. These algebras are determined combinatorially by mutation classes of certain rectangular matrices. It was conjectured, by Fomin and Zelevinsky, that finite type cluster algebras with principal coefficients are characterized by the mutation classes which are finite. In this paper, we prove this conjecture. [Copyright &y& Elsevier]

Abstract: Let be a graph with vertex set V(G) and edge set E(G). The subdivision graph of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let and be two vertex disjoint graphs. The subdivision-vertex neighbourhood corona of and , denoted by , is the graph obtained from and copies of , all vertex disjoint, and joining the neighbours of the ith vertex of to every vertex in the ith copy of . The subdivision-edge neighbourhood corona of and , denoted by , is the graph obtained from and copies of , all vertex disjoint, and joining the neighbours of the ith vertex of to every vertex in the ith copy of , where is the set of inserted vertices of . In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of (respectively, ) in terms of the corresponding spectra of and . As applications, these results enable us to construct infinitely many pairs of cospectral graphs, and using the results on the Laplacian spectra of subdivision-vertex neighbourhood coronae, new families of expander graphs are constructed from known ones. [Copyright &y& Elsevier]

Abstract: An n × n matrix A has a normal defect of k if there exists an normal matrix with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4×4 matrices, namely matrices whose only nonzero entries lie on the superdiagonal, and we provide details for constructing minimal normal completion matrices . We also prove a result for a related class of n × n matrices. Finally, we present an example of a 6×6 block diagonal matrix having the property that its normal defect is strictly less than the sum of the normal defects of each of its blocks, and we provide sufficient conditions for when the normal defect of a block diagonal matrix is equal to the sum of the normal defects of each of its blocks. [Copyright &y& Elsevier]

Abstract: Crouzeix’s conjecture is that for any square matrix A and any polynomial p we havewhere is the field of values of A and denotes the spectral norm. In this paper, we show that the conjecture holds for the matrices of the formwhere and . [Copyright &y& Elsevier]

Abstract: For two n-by-n matrices, A,B, the product field of values is the set . In this paper, we establish basic properties of the product field of values. The main results are a proof that the product field is a simply connected subset of the complex plane and a characterization of matrix pairs for which the product field has nonempty interior. [Copyright &y& Elsevier]

Abstract: In this paper we present the classification of a subclass of naturally graded Leibniz algebras. This subclass has the nilindex and the characteristic sequence where n is the dimension of the algebra. In fact, this result completes the classification of naturally graded Leibniz algebras of nilindex . [Copyright &y& Elsevier]

Abstract: This paper deals with nonnegativity of matrices and their group or Drazin inverses. Firstly, the nonnegativity of a square matrix A, its group inverse and its group projector is used to define different sets for which relationships and characterizations are given. Next, an extension of the previous results for index greater than 1 is presented. Similar sets are introduced and studied for Drazin inverses and Drazin projectors considering the core-nilpotent decomposition. In addition, the results are applied to study the -Drazin periodic matrices for . [Copyright &y& Elsevier]

Abstract: In this paper, -potent matrices are considered. A matrix is called -potent when where K is an involutory matrix and . Specifically, -potent matrices are analyzed considering their relations to different classes of complex matrices. These classes of matrices are: -generalized projectors, -Hermitian matrices, normal matrices, and matrices (anti-)commuting with K or such that KB is involutory, Hermitian or normal. In addition, some new relations for K-generalized centrosymmetric matrices have been derived. [Copyright &y& Elsevier]

Abstract: In various application fields, tensor type data are used recently and then a typical rank is important. There may be more than one typical ranks over the real number field. It is well known that the set of tensors has two typical ranks for , that the set of tensors has two typical ranks , and that the set of tensors has two typical ranks . In this paper, we show that the set of tensors with has two typical ranks if , where is the Hurwitz-Radon function defined as for nonnegative integers such that and . [Copyright &y& Elsevier]

Abstract: Let G σ be an oriented graph obtained by assigning an orientation σ to the edge set of a simple undirected graph G such that G σ becomes a directed graph. Let S(G σ ) be the skew adjacency matrix of G σ . The skew energy of G σ is defined as the sum of the absolute values of all eigenvalues of S(G σ ). In this paper, we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first largest skew energies for . [Copyright &y& Elsevier]

Abstract: Let be the set of all bounded linear operators on a Hilbert space . In this paper we show that if S is a closed range operator with , then if and only if S is a non-zero real scalar of a normal partial isometry. Also we find some other characterizations of this space using some inequalities related to Corach–Porta–Recht inequality. [Copyright &y& Elsevier]

Abstract: Unicyclic graphs are connected graphs in which the number of edges equals the number of vertices. Let be the class of unicyclic graphs of order n and diameter d. For unicyclic graphs, lexicographic ordering by spectral moments (S-order) is discussed in this paper. The last graphs, in the S-order, among all unicyclic graphs in are characterized. In addition, for the cases and , the last and last unicyclic graphs in the set are also given, respectively. [Copyright &y& Elsevier]

Abstract: Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions. [Copyright &y& Elsevier]

Abstract: In this paper we use doubly stochastic operators to extend the notion of majorization on , where I is assumed to be an infinite set and , and then characterize the structure of all bounded linear maps on this space which preserve majorization. We will see that the two cases and lead to different structures. [Copyright &y& Elsevier]

Abstract: In this paper, for a given set of numbers with special conditions, we construct a nonnegative matrix such that the given set is its spectrum. [Copyright &y& Elsevier]

Abstract: Let be the set of all the integer points in the m-dilate of the Birkhoff polytope of doubly-stochastic matrices. In this paper we find the sharp upper bound on the tropical determinant over the set . We define a version of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum and then find the sharp lower bound on thus defined tropical determinant over . [Copyright &y& Elsevier]

Abstract: Two matrices , are called thin Lehman matrices if they are solutions of the matrix equation , where is the matrix of all s and is the identity matrix. These matrices are important in the set covering problem, but few examples are known. In this paper, we will introduce the notion of -overlapped factorizations of finite groups which constructs a new class of thin Lehman matrices. Moreover, we will study some structural properties of -overlapped factorizations. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the complete positivity of house matrices, i.e., doubly nonnegative matrices whose graph is a cycle with a chord between vertices 1 and 3. We describe the set of all possible supports of the columns of a nonnegative matrix B,such that; show that the cp-rank of a completely positive house matrix is at least ; give a complete characterization of the singular completely positive house matrices; show that their cp-rank is or n and characterize each case. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the ordering on a semiring of monotone doubly stochastic transition matrices in Shorrocks’ sense. We identify a class of an equilibrium index of mobility that induces the full ordering in a semiring, while this ordering is compatible with Dardanoni’s partial ordering on a set of monotone primitive irreducible doubly stochastic matrices. [Copyright &y& Elsevier]

Abstract: Let be the Schatten p-class for . Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if and are two sets of operators in, then In this paper, we give generalizations of this as pairs of inequalities for Schatten p-norms, which hold for certain values of p and reduce to the equality above for . Moreover, we present some related inequalities for three sets of operators. [Copyright &y& Elsevier]

Abstract: The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. Bao, Tan and Fan [Y.H. Bao, Y.Y. Tan,Y.Z. Fan, The Laplacian spread of unicyclic graphs, Appl. Math. Lett. 22 (2009) 1011–1015.] characterize the unique unicyclic graph with maximum Laplacian spread among all connected unicyclic graphs of fixed order. In this paper, we characterize the unique quasi-tree graph with maximum Laplacian spread among all quasi-tree graphs in the set with . [Copyright &y& Elsevier]

Abstract: We construct a class of boolean invertible matrices whose elements satisfy the following property: when we perform the Hadamard product operation on the set of row vectors of an element we produce either the row or the zero row. In this paper, we prove that every matrix is uniquely determined by a pair of permutations of the set . As a by-product of this result we identify Haar-type matrices from a pair of permutations as well, because these matrices emerge from the Gram–Schmidt orthonormalization process of the set of row vectors of R matrices belonging in a certain subclass . [Copyright &y& Elsevier]

Abstract: Let be a multiplicative semigroup of matrices with nonnegative entries. Assume that the diagonal entries of the members of form a finite set. This paper is concerned with the following question: Under what circumstances can we deduce that itself is finite? [Copyright &y& Elsevier]

Abstract: In this paper, we compute the natural density of the set of integer matrices that can be extended to an invertible matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form . Connections with Cesàro’s Theorem on the density of coprime integers and Quillen–Suslin’s Theorem are also presented. [ABSTRACT FROM AUTHOR]

Abstract: A sign pattern is said to be nilpotent of index k if all real matrices in its qualitative class are nilpotent and their maximum nilpotent index equals k. In this paper, we characterize sign patterns that are nilpotent of a given index k. The maximum number of nonzero entries in such sign patterns of a given order is determined as well as the sign patterns with this maximum number of nonzero entries. [Copyright &y& Elsevier]

Abstract: quiangular frames are an important class of finite dimensional frames because of their superior performance and numerous applications. The objective of this paper is to present a new tool to construct equiangular tight frames from groups. We prove that many equiangular tight frames arise from subsets of groups which we call signature sets. Subsequently, we define quasi-signature sets and examine real equiangular tight frames associated with these subsets of groups. This approach yields further results and establishes new correspondences. We extend these results to complex equiangular tight frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular tight frames that arise from groups in this manner. [ABSTRACT FROM AUTHOR]

Abstract: Let be the algebra of all bounded linear operators on a complex Hilbert space In 1978, Drazin introduced a partial order, on For if and only if and Given let be the least upper bound (supremum) for A and B with respect to the ∗-order. In this paper, we present some necessary and sufficient conditions for which exists, and give an explicit representation of (if exists). [ABSTRACT FROM AUTHOR]

Abstract: Let be the class of trees with vertices and diameter . In this paper, the lexicographic ordering of trees in the set by spectral moments is considered, and the last trees, in an -order, among all trees in are characterized. Moreover, all trees in have an -order for and . [ABSTRACT FROM AUTHOR]

Abstract: Böttcher and Wenzel recently proved that for any unitarily invariant norm , and that when the norm is the Frobenius norm. They also asked whether the Frobenius norm is the only one having such property. In this paper, we answer the question by showing that the dual norm of the -norm also has the property that . [ABSTRACT FROM AUTHOR]

Abstract: Let be an interval matrix and . We introduce the concept of Schur and Hurwitz diagonal stability, relative to the Hölder p-norm, of , abbreviated as and , respectively. This concept is formulated in terms of a matrix inequality using the p-norm, which must be satisfied by the same positive definite diagonal matrix for all . The inequality form is different for and . The particular case of is equivalent to the condition of quadratic stability of . The inequality is equivalent to the Stein inequality , and the inequality is equivalent to the Lyapunov inequality ; in both cases P is a positive definite diagonal matrix and the notation “” means negative definite. The first part of the paper provides and criteria, presents methods for finding the positive definite diagonal matrix requested by the definition of and , analyzes the robustness of and and explores the connection with the Schur and Hurwitz stability of . The second part shows that the or of is equivalent to the following properties of a discrete- or continuous-time dynamical interval system whose motion is described by : the existence of a strong Lyapunov function defined by the p-norm and the existence of exponentially decreasing sets defined by the p-norm that are invariant with respect to system’s trajectories. [Copyright &y& Elsevier]

Abstract: In this paper we establish algebraic reflexivity properties of subsets of bounded linear operators acting on spaces of vector valued Lipschitz functions. We also derive a representation for the generalized bi-circular projections on these spaces. [Copyright &y& Elsevier]

Abstract: A subset of the vertex set of a graph , , is a -regular set if it induces a -regular subgraph of and every vertex not in the subset has neighbors in it. In this paper we deal with the existence of -regular sets associated with all distinct eigenvalues. We show some families that have this property and we give some results concerning the existence of such sets considering restrictions on the symbol of circulant graphs. [Copyright &y& Elsevier]

Abstract: For positive integers and satisfying and , let be the bipartite graph with partite sets and such that and are not adjacent if and only if there exists a positive integer with such that and . In this paper we study the largest eigenvalues of bipartite graphs which are nearly complete. We first compute the largest eigenvalue (and all other eigenvalues) of , and then list nearly complete bipartite graphs according to the magnitudes of their largest eigenvalues. These results give an affirmative answer to [1, Conjecture 1.2] when the number of edges of a bipartite graph with partite sets and , having and for , is . Furthermore, we refine [1, Conjecture 1.2] for the case when the number of edges is at least . [Copyright &y& Elsevier]