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We determine the region of singular values of the commutator X Y − Y X when X and Y vary over the set of norm one rank one matrices. This answers in affirmative a conjecture raised in Wenzel (2015) [7]. [ABSTRACT FROM AUTHOR]

Abstract We consider the problem of updating the SVD when augmenting a "tall thin" matrix, i.e., a rectangular matrix A ∈ R m × n with m ≫ n. Supposing that an SVD of A is already known, and given a matrix B ∈ R m × n ′ , we derive an efficient method to compute and efficiently store the SVD of the augmented matrix [ A B ] ∈ R m × (n + n ′). This is an important tool for two types of applications: in the context of principal component analysis, the dominant left singular vectors provided by this decomposition form an orthonormal basis for the best linear subspace of a given dimension, while from the right singular vectors one can extract an orthonormal basis of the kernel of the matrix. We also describe two concrete applications of these concepts which motivated the development of our method and to which it is very well adapted. [ABSTRACT FROM AUTHOR]

It is proved that regularity of an interval matrix implies singularity of four related interval matrices. The result is used to prove that for each nonsingular point matrix A , either A or A − 1 can be brought to a singular matrix by perturbing only the diagonal entries by an amount of at most 1 each. As a consequence, the notion of a diagonally singularizable matrix is introduced. [ABSTRACT FROM AUTHOR]

We prove a result concerning the behavior of the spectral radius of a hypergraph under relocations of edges. We determine the unique hypergraphs with maximum spectral radius among connected k -uniform hypergraphs with fixed number of pendant edges, the unique k -uniform hypertrees with respectively maximum, second maximum and third maximum spectral radius, the unique k -uniform unicyclic hypergraphs ( k -uniform linear unicyclic hypergraphs, respectively) with respectively maximum and second maximum spectral radius. We also determine the unique hypergraphs with maximum spectral radius among k -uniform unicyclic hypergraphs with given girth. [ABSTRACT FROM AUTHOR]

W.E. Roth (1952) proved that the matrix equation A X − X B = C has a solution if and only if the matrices [ A C 0 B ] and [ A 0 0 B ] are similar. A. Dmytryshyn and B. Kågström (2015) extended Roth's criterion to systems of matrix equations A i X i ′ M i − N i X i ″ σ i B i = C i ( i = 1 , … , s ) with unknown matrices X 1 , … , X t , in which every X σ is X , X ⊤ , or X ⁎ . We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations. [ABSTRACT FROM AUTHOR]

The columns of an m × n ACI-matrix over a field F are independent affine subspaces of F m . An ACI-matrix has constant rank ρ if all its completions have rank ρ . Huang and Zhan (2011) [4] characterized the m × n ACI-matrices of constant rank when | F | ≥ min ⁡ { m , n + 1 } . We complete their result characterizing the m × n ACI-matrices of constant rank over arbitrary fields. Quinlan and McTigue (2014) [8] proved that every partial matrix of constant rank ρ has a ρ × ρ submatrix of constant rank ρ if and only | F | ≥ ρ . We obtain an analogous result for ACI-matrices over arbitrary fields by introducing the concept of complete irreducibility. [ABSTRACT FROM AUTHOR]

It seems to be of recurring interest in the literature to give alternative proofs for the fact that the determinant of a symplectic matrix is one. We state four short and elementary proofs for symplectic matrices over general fields. Two of them seem to be new. [ABSTRACT FROM AUTHOR]

A correlation matrix is a positive semi-definite Hermitian matrix with all diagonals equal to 1. The minimum of the permanents on singular correlation matrices is conjectured to be given by the matrix Y n , all of whose non-diagonal entries are − 1 / ( n − 1 ) . Also, Frenzen–Fischer proved that per Y n approaches to e / 2 as n → ∞ . In this paper, we analyze some immanants of Y n , which are the generalizations of the determinant and the permanent, and we generalize these results to some other immanants and conjecture most of those converge to 1. [ABSTRACT FROM AUTHOR]

The matrix equation A X − X B = C has a solution if and only if the matrices [ A C 0 B ] and [ A 0 0 B ] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X − A X B = C over a field has a solution if and only if the matrices [ A C 0 I ] and [ I 0 0 B ] are simultaneously equivalent to [ A 0 0 I ] and [ I 0 0 B ] . We extend these criteria to the matrix equations A X − X ˆ B = C and X − A X ˆ B = C over the skew field of quaternions with a fixed involutive automorphism q ↦ q ˆ . [ABSTRACT FROM AUTHOR]

For a matrix A whose off-diagonal entries are nonpositive, its nonnegative invertibility (namely, that A is an invertible M -matrix) is equivalent to A being a P -matrix, which is necessary and sufficient for the unique solvability of the linear complementarity problem defined by A . This, in turn, is equivalent to the statement that A is strictly semimonotone. In this paper, an analogue of this result is proved for singular symmetric Z -matrices. This is achieved by replacing the inverse of A by the group generalized inverse and by introducing the matrix classes of strictly range semimonotonicity and range column sufficiency. A recently proposed idea of P # -matrices plays a pivotal role. Some interconnections between these matrix classes are also obtained. [ABSTRACT FROM AUTHOR]

An n × n complex matrix A is called coninvolutory if A ¯ A = I n and skew-coninvolutory if A ¯ A = − I n (which implies that n is even). We prove that each matrix of size n × n with n > 1 is a sum of 5 coninvolutory matrices and each matrix of size 2 m × 2 m is a sum of 5 skew-coninvolutory matrices. We also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix. A matrix M is called condiagonalizable if M = S ¯ − 1 D S in which S is nonsingular and D is diagonal. [ABSTRACT FROM AUTHOR]

Yuan and Leng (2007) gave a generalization of the matrix form of the Brunn–Minkowski inequality. In this note, we first give a simple proof of this inequality, and then show a generalization of this to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. [ABSTRACT FROM AUTHOR]

We show that the d -cube is determined by the spectrum of its distance matrix. [ABSTRACT FROM AUTHOR]

For any algebraically closed field F and any two square matrices A , B over F , Sylvester (1884) [8] and Cecioni (1910) [1] showed that A X = X B implies X = 0 if and only if A and B have no common eigenvalue. It is proved that a third equivalent statement is that, for any given polynomials f , g in F [ t ] , there exists h in F [ t ] such that f ( A ) = h ( A ) and g ( B ) = h ( B ) . Corresponding results hold also for any finite set of square matrices over F , and these lead to a new property of all associative rings and algebras (even over arbitrary fields) with 1. [ABSTRACT FROM AUTHOR]

A method for generating all rational generalized matrices on indefinite real inner product spaces isomorphic to R p , q is presented. The proposed method is based on the proof of a weak version of the Cartan–Dieudonné theorem, handled using Clifford algebras. It is shown that all rational B -orthogonal matrices in an indefinite inner product space ( X , B ) are products of simple matrices with rational entries. [ABSTRACT FROM AUTHOR]

Let R be a unital associative algebra over a field K of characteristic zero, and let f be a multilinear polynomial of degree m over K . If m ≤ 3 , we prove that all traceless matrices can be written as the sum of two values of f evaluated over M n ( R ) with n ≥ 2 . If m = 4 , we prove the same result for n ≥ 3 . [ABSTRACT FROM AUTHOR]

A matrix A ∈ M n ( R ) with coefficients in any ring R is a quasi-permutation matrix if each row and each column has at most one nonzero element. It is shown that a singular quasi-permutation matrix with coefficients in a domain is a product of idempotent matrices. As an application, we prove that a nonnegative singular matrix having nonnegative von Neumann inverse (also known as generalized inverse) is a product of nonnegative idempotent matrices. [ABSTRACT FROM AUTHOR]

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type Tr f ( Φ ( A p ) 1 / 2 Ψ ( B q ) Φ ( A p ) 1 / 2 ) and symmetric (anti-) norm functions of the form ‖ f ( Φ ( A p ) σ Ψ ( B q ) ) ‖ , where Φ and Ψ are positive linear maps, σ is an operator mean, and f ( x γ ) with a certain power γ is an operator monotone function on ( 0 , ∞ ) . Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on ( 0 , ∞ ) so that we prove joint concavity/convexity of more trace functions of Lieb type. [ABSTRACT FROM AUTHOR]

In this paper, we explore various matrix-theoretic aspects of the hidden Z class and demonstrate how the concept of principal pivot transform can be effectively used to extend many existing results. In fact, we revisit various results obtained for hidden Z class by Mangasarian, Cottle and Pang in context of solving linear complementarity problems as linear programs. We identify hidden Z -matrices of special category and discuss the number of solutions of the associated linear complementarity problems. We also present game theoretic interpretation of various results related to hidden Z class and obtain proofs following the game theoretic approach of Raghavan for a subclass of Z -matrices. [ABSTRACT FROM AUTHOR]

In this paper we shall study non-commutative central moment inequalities with a focus on whether the commutative bounds are tight in the non-commutative case as well. We prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case. As an application, we shall present some lower estimates of the spread of Hermitian and normal matrices as well. [ABSTRACT FROM AUTHOR]

In this paper, we reformulate the nonlinear complementarity problem as an implicit fixed-point equation. We establish a modulus-based matrix splitting iteration method based on the implicit fixed-point equation and prove its convergence theorem under suitable conditions. Furthermore, we propose a two-step modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. We can obtain many matrix splitting iteration methods by suitably choosing the matrix splittings and the parameters. The proposed methods can be regarded as extensions of the methods for linear complementarity problem. Numerical experiments are presented to show the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

We call an n -tuple Q 1 , … , Q n of positive definite n × n real matrices α -conditioned for some α ≥ 1 if for the corresponding quadratic forms q i : R n ⟶ R we have q i ( x ) ≤ α q i ( y ) for any two vectors x , y ∈ R n of Euclidean unit length and q i ( x ) ≤ α q j ( x ) for all 1 ≤ i , j ≤ n and all x ∈ R n . An n -tuple is called doubly stochastic if the sum of Q i is the identity matrix and the trace of each Q i is 1. We prove that for any fixed α ≥ 1 the mixed discriminant of an α -conditioned doubly stochastic n -tuple is n O ( 1 ) e − n . As a corollary, for any α ≥ 1 fixed in advance, we obtain a polynomial time algorithm approximating the mixed discriminant of an α -conditioned n -tuple within a polynomial in n factor. [ABSTRACT FROM AUTHOR]

Let K n be the set of all n × n lower triangular ( 0 , 1 ) -matrices with each diagonal element equal to 1, L n = { Y Y T : Y ∈ K n } and let c n be the minimum of the smallest eigenvalue of Y Y T as Y goes through K n . The Ilmonen–Haukkanen–Merikoski conjecture (the IHM conjecture) states that c n is equal to the smallest eigenvalue of Y 0 Y 0 T , where Y 0 ∈ K n with ( Y 0 ) i j = 1 − ( − 1 ) i + j 2 for i > j . In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices. [ABSTRACT FROM AUTHOR]

In this note, we determined the distance signatures of the incidence matrices of affine resolvable designs. This proves a conjecture by Kohei Yamada. [ABSTRACT FROM AUTHOR]

In this paper we prove a new characterization of the max-plus singular values of a max-plus matrix, as the max-plus eigenvalues of an associated max-plus matrix pencil. This new characterization allows us to compute max-plus singular values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially parameterized we show experimentally that max-plus singular values give order of magnitude approximations to the classical singular values of parameter independent classical matrices. We also discuss Hungarian scaling, which is a diagonal scaling strategy for preprocessing classical linear systems. We show that Hungarian scaling can dramatically reduce the 2-norm condition number and that this action can be explained using our new theory for max-plus singular values. [ABSTRACT FROM AUTHOR]

E.M. Sá and R.C. Thompson proved that the invariant factors of a matrix over a commutative principal ideal domain and the invariant factors of its submatrices are related by a set of divisibility inequalities, called the interlacing inequalities for invariant factors. We extend this result to matrices over elementary divisor duo rings. [ABSTRACT FROM AUTHOR]

We investigate the operator monotonicity of the following functions: f ( t ) = t γ ( t α 1 − 1 ) ( t α 2 − 1 ) ⋯ ( t α n − 1 ) ( t β 1 − 1 ) ( t β 2 − 1 ) ⋯ ( t β n − 1 ) ( t ∈ ( 0 , ∞ ) ) , where γ ∈ R and α i , β j > 0 with α i ≠ β j ( i , j = 1 , 2 , … , n ). This property for these functions has been considered by V.E.S. Szabó [14] . [ABSTRACT FROM AUTHOR]

We prove inequalities on symmetric tensor sums of positive definite operators. In particular, we prove multivariable operator inequalities inspired by generalizations to the well-known Hlawka and Popoviciu inequalities. As corollaries, we obtain generalized Hlawka and Popoviciu inequalities for determinants, permanents, and generalized matrix functions. The new operator inequalities and their corollaries contain a few recently published inequalities on positive definite matrices as special cases. [ABSTRACT FROM AUTHOR]

We use operator monotone and operator convex functions to prove an inverse to the Young inequality for eigenvalues of positive definite matrices and then apply it to obtain a matrix inverse Young inequality which can be considered as a complement of a result of T. Ando. Also, we give a necessary and sufficient condition for the equality. [ABSTRACT FROM AUTHOR]

We give new proofs of several fundamental results of spectral theory in max algebra. This includes the description of the spectrum in max algebra of a given non-negative matrix via local spectral radii, the spectral theorem and the spectral mapping theorem in max algebra. The latter result is also generalized to the setting of power series in max algebra by applying certain continuity properties of the spectrum in max algebra. Our methods enable us to obtain some related results for the usual spectrum of complex matrices and distinguished spectrum for non-negative matrices. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of maximizing a quadratic form 〈 A x , x 〉 subject to ‖ x ‖ q = 1 , where A has matrix entries f ( [ i , j ] ( i , j ) ) with i , j | k and q ≥ 1 . We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x 1 x m n = x m x n . This turns out to depend on both f and q , with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f multiplicative such that 0 < f ( p ) < 1 , the solution is at a multiplicative point for all q ≥ 1 . [ABSTRACT FROM AUTHOR]

The distance matrix of a simple graph G is D ( G ) = ( d i j ) , where d i j is the distance between i th and j th vertices of G . The spectrum of the distance matrix is known as the distance spectrum or D -spectrum of G . A simple connected graph G is called distance regular if it is regular, and if for any two vertices x , y ∈ G at distance i , there are constant number of neighbors c i and b i of y at distance i − 1 and i + 1 from x respectively. In this paper we prove that distance regular graphs with diameter d have at most d + 1 distinct D -eigenvalues. We find an equitable partition and the corresponding quotient matrix of the distance regular graph which gives the distinct D -eigenvalues. We also prove that distance regular graphs satisfying b i = c d − 1 have at most ⌈ d 2 ⌉ + 2 distinct D -eigenvalues. Applying these results we find the distance spectrum of some distance regular graphs including the well known Johnson graphs. Finally we also answer the questions asked by Lin et al. [16] . [ABSTRACT FROM AUTHOR]

From the parameters ( n , k , t , λ , μ ) of a directed strongly regular graph (dsrg) A. Duval (1988) [4] showed how to compute the eigenvalues and multiplicities of the adjacency matrix, and thus the rank of the adjacency matrix. For every rational number q , where 1 5 ≤ q ≤ 7 10 , there is a feasible (i.e., satisfying Duval's conditions) parameter set for a dsrg with rank 5 and with k n = q . In this paper we show that there exist a dsrg with such a feasible parameter set only if k n is 1 5 , 1 3 , 2 5 , 1 2 , 3 5 , or 2 3 . Every dsrg with rank 5 therefore has parameters of a known graph. The proof is based on an enumeration of 5 × 5 matrices with entries in { 0 , 1 } . [ABSTRACT FROM AUTHOR]

A P-set of a symmetric matrix A is a set α of indices such that the nullity of the matrix obtained from A by removing rows and columns indexed by α is | α | more than the nullity of A . It is known that each subset of a P-set is a P-set. It is also known that a set of indices such that each singleton subset is a P-set need not be a P-set. This note shows that if all pairs of vertices of a set with at least two elements are P-sets, then the set is a P-set. [ABSTRACT FROM AUTHOR]

In this paper, an explicit formula for the ranks of dense alternating sign matrices is obtained. The minimum rank and the maximum rank of the sign pattern of a dense alternating sign matrix are determined. Some related results and examples are also provided. [ABSTRACT FROM AUTHOR]

Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into two diagonal unitaries and one whose row- and column sums are equal to one. The proof is non-constructive and based on a reformulation in terms of symplectic topology. As a corollary, we obtain a decomposition of unitary matrices into an interlaced product of unitary diagonal matrices and discrete Fourier transformations. This provides a new decomposition of linear optics arrays into phase shifters and canonical multiports described by Fourier transformations. [ABSTRACT FROM AUTHOR]

Hamiache assigns to every TU game a so-called associated game and then shows that the Shapley value is characterized as the unique solution for TU games satisfying the inessential game property, continuity and associated consistency. The latter notion means that for every game the Shapley value of the associated game is equal to the Shapley value of the game itself. In this paper we show that also the EANS-value as well as the CIS-value is characterized by these three properties for appropriately modified notions of the associated game. This shows that these three values only differ with respect to the associated game. The characterization is obtained by applying the matrix approach as the pivotal technique for characterizing linear values of TU games in terms of associated consistency. [ABSTRACT FROM AUTHOR]

We first prove a matrix identity concerning the blocks of generalized Jordan blocks and then give applications to some invariants of matrices. As a consequence, we reprove the well known fact that for an eigenvalue λ , its algebraic multiplicity is greater than or equal to its geometric multiplicity. [ABSTRACT FROM AUTHOR]

Ray nonsingular (RNS) matrices are a generalization of sign nonsingular (SNS) matrices from the real field to the complex field. A matrix with positive diagonals is called a cycle tree matrix if the adjacency structure of all the cycles in its associated digraph is a tree. In this paper, the ray nonsingularity of cycle tree matrices is studied, and a general method is given to recognize ray nonsingular cycle tree matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of Lie algebras which have free t-nilpotent Lie algebras n2,t of type 2 as nilradical and give a detailed construction for them. We prove that the dimension of any Lie algebra g of this class is dimn2,t+k. If g is solvable, k≤2; otherwise, the Levi subalgebra of g is sl2(K), the split simple 3-dimensional Lie algebra of 2×2 matrices of trace zero, and then k≤4. As an application of the main results we get the classification over algebraically closed fields of Lie algebras with nilradical n2,1, n2,2 and n2,3. [ABSTRACT FROM AUTHOR]

It is easily checkable if a given tensor is a B tensor, or a B0 tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B0 tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B0 tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case. [ABSTRACT FROM AUTHOR]

The optimal preconditioner cU(A) of a given matrix A was proposed in 1988 by T. Chan [6]. Since then, it has been proved to be efficient for solving a large class of structured systems. In this paper, we construct the optimal preconditioners for different functions of matrices. More precisely, let f be a function of matrices from Cn×n to Cn×n. Given A∈Cn×n, there are two possible optimal preconditioners for f(A): cU(f(A)) and f(cU(A)). In the paper, we study properties of both cU(f(A)) and f(cU(A)) for different functions of matrices. Numerical experiments are given to illustrate the efficiency of the optimal preconditioners when they are used to solve f(A)x=b. [ABSTRACT FROM AUTHOR]

One of fundamental results of the theory of joint/generalized spectral radius, the Berger-Wang theorem, establishes equal-ity between the joint and generalized spectral radii of a set of matrices. Generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with essential difficulties since known proofs of the Berger-Wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products. Recently, X. Dai [1] proved an analog of the Berger-Wang theorem for the case when factors in matrix products are formed by some Markov law. We extend the concepts of joint and generalized spectral radii to products of matrices with constraints on the sliding block relative frequencies of occurrences of different factors, and prove an analog of the Berger-Wang theorem for this case. [ABSTRACT FROM AUTHOR]

Abstract: A continuous-time quantum walk on a graph G is given by the unitary matrix , where A is the adjacency matrix of G. We say G has pretty good state transfer between vertices a and b if for any , there is a time t, where the -entry of satisfies . This notion was introduced by Godsil (2011). The state transfer is perfect if the above holds for . In this work, we study a natural extension of this notion called universal state transfer wherein state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property: (i) Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (each eigenvector has entries which are equal in magnitude). (ii) The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, the switching automorphism group is cyclic. (iii) There is a family of complex oriented prime-length cycles which has universal pretty good state transfer. This provides a concrete example of a family of graphs with this universal property. (iv) There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer. Finally, we provide a spectral characterization of universal perfect state transfer for graphs switching equivalent to circulants. [Copyright &y& Elsevier]

In this expository paper, we analyze and compare two determinantal identities constructed in the mid nineteenth century, Chiò's and Dodgson's, from the perspective of their origins in earlier work by J.J. Sylvester and C.G.J. Jacobi. All the known proofs of Chiò's and Dodgson's identities are cited and we present and compare them as modern algorithms for evaluating determinants. Restated as a recurrence, we discuss the role of Dodgson's identity in the development of the alternating sign matrix conjecture in the late twentieth century. [ABSTRACT FROM AUTHOR]

Abstract: We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) matrix is (resp. ). Relationships with point-line incidences in the plane, Bruhat order of permutations, and TP completability are also presented. We also examine the number and positionings of equal minors in a TP matrix, and give a relationship between the location of equal minors and outerplanar graphs. [Copyright &y& Elsevier]

Abstract: In this paper we will present a new proof of the main theorem in Linear Algebra Appl. 439 (2013) 4063-4069. [Copyright &y& Elsevier]

Abstract: Let be the variety of all d-tuples of commuting nilpotent matrices. It is well-known that is irreducible if , if or if and . On the other hand is known to be reducible for . We study in this paper the reducibility of for various values of d and n. In particular, we prove that is reducible for all . In the case , we show that it is irreducible for . [Copyright &y& Elsevier]

Abstract: For a distance matrix , its inertia is the triple of integers , where , , denote the number of positive, 0, negative eigenvalues of , respectively. The D-energy is the sum of the absolute eigenvalues of . In this paper, we first study the inertia of distance matrices of complete k-partite graphs; Then, as applications, we not only prove a conjecture proposed by Caporossi et al. (2009) in [5] in a different way from Stevanović et al. (2013) [11] but also obtain the formula of the D-energy of the remaining complete k-partite graphs. At last, we obtain the graphs with the maximum (resp. minimum) D-energy among all the complete k-partite graphs with n vertices. [Copyright &y& Elsevier]

Abstract: The problem of determining whether or not an integer matrix is diagonally similar to an integer matrix with off-diagonal entries equal to 1 is studied. Such a matrix is called integrally normalizable, and a zero–nonzero pattern is integrally normalizable if each matrix with this zero–nonzero pattern is integrally normalizable with respect to the same set of off-diagonal entries. Matrices that are integrally normalizable with respect to a fixed spanning tree, and integrally normalizable zero–nonzero patterns are characterized. The maximum number of nonzero entries in an integrally normalizable zero–nonzero pattern is shown to be . Extensions to matrices over other integral domains are also presented. [Copyright &y& Elsevier]