Local bases of primitive non-powerful signed digraphs

In 1994, Z. Li, F. Hall and C. Eschenbach extended the concept of the index of convergence from nonnegative matrices to powerful sign pattern matrices. Recently, Jiayu Shao and Lihua You studied the bases of non-powerful irreducible sign pattern matrices. In this paper, the local bases, which are generalizations of the base, of primitive non-powerful signed digraphs are introduced, and sharp bounds for local bases of primitive non-powerful signed digraphs are obtained. Furthermore, extremal digraphs are described.     

Bounds on the kth multi-g base index of nearly reducible sign pattern matrices

Li et al. [On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101-120.] extended the concepts of the base and period from nonnegative matrices to powerful sign pattern matrices. Then, Shao and You [Bound on the basis of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 285-300.] extended the concepts of the base from powerful sign pattern matrices to non-powerful irreducible sign pattern matrices. In this paper we mainly study the kth multi-g base index for non-powerful primitive nearly reducible sign pattern matrices. We obtain sharp upper bounds, together with a complete characterization of the equality cases of the kth multi-g base index for primitive nearly reducible generalized sign pattern matrices. We also show that there exist “gaps” in the kth multi-g base index set of the classes of such matrices.     

The kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs

In this article, we study the kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs. A bound on the kth upper bases for primitive nonpowerful minimally strong signed digraphs is obtained, and the equality case of the bound is characterized. For the kth lower bases, we obtain some bounds. For some cases, the bounds are best possible and the extremal signed digraphs are characterized. We also show that there exist ‘gaps’ in both the kth upper base set and the kth lower base set of primitive nonpowerful minimally strong signed digraphs.     

The Generalized <Emphasis Type="Italic">τ</Emphasis>-Bases of Primitive Non-Powerful Signed Digraphs with d Loops

Let n, k, τ, d be positive integers with 1 ≤ k, τ, d ≤ n. As natural extensions of the bases, the kth local bases, the kth upper bases and the kth lower bases of primitive non-powerful signed digraphs, we introduce a number of new, though, intimately related parameters called the generalized τ-bases of primitive non-powerful signed digraphs. Moreover, some sharp bounds for the generalized τ-bases of primitive non-powerful signed digraphs with n vertices and d loops are obtained, respectively.     

The base sets of primitive zero-symmetric sign pattern matrices

In [J.Y. Shao, L.H. You, Bound on the base of irreducible generalized sign pattern matrices, Discrete Math., in press], Shao and You extended the concept of the base from powerful sign pattern matrices to non-powerful (and generalized) sign pattern matrices. In this paper, we study the bases of primitive zero-symmetric sign pattern (and generalized sign pattern) matrices. Sharp upper bounds of the bases are obtained. We also show that there exist no “gaps” in the base sets of the classes of such matrices.     

Bounds on the local bases of primitive nonpowerful nearly reducible sign patterns

In this work, we study the kth local base, which is a generalization of the base, of a primitive non-powerful nearly reducible sign pattern of order n ≥ 7. We obtain the sharp bound together with a complete characterization of the equality case, of the kth local bases for primitive non-powerful nearly reducible sign patterns. We also show that there exist “gaps” in the kth local base set of primitive non-powerful nearly reducible sign patterns.     

Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs

In [B.M. Kim, B.C. Song, W. Hwang, Primitive graphs with given exponents and minimum number of edges, Linear Algebra Appl. 420 (2007) 648-662], the minimum number of edges of a simple graph on n vertices with exponent k was determined. In this paper, we completely determine the minimum number, H(n,k), of arcs of primitive non-powerful symmetric loop-free signed digraphs on n vertices with base k, characterize the underlying digraphs which have H(n,k) arcs when k is 2, nearly characterize the case when k is 3 and propose an open problem.     

Bounds on the k-th generalized base of a primitive sign pattern matrix

You et al. [L. You, J. Shao, and H. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Lin. Alg. Appl. 427 (2007), pp. 285–300] extended the concept of the base of a powerful sign pattern matrix to the nonpowerful, irreducible sign pattern matrices. The key to their generalization was to view the relationship A ^l =A ^l?+?p as an equality of generalized sign patterns rather than of sign patterns. You, Shao and Shan showed that for primitive generalized sign patterns, the base is the smallest positive integer k such that all entries of A ^k are ambiguous. In this paper we study the k-th generalized base for nonpowerful primitive sign pattern matrices. For a primitive, nonpowerful sign pattern A, this is the smallest positive integer h such that A^k has h rows consisting entirely of ambiguous entries. Extending the work of You, Shao and Shan, we obtain sharp upper bounds on the k-th generalized base, together with a complete characterization of the equality cases for those bounds. We also show that there exist gaps in the k-th generalized base set of the classes of such matrices.     

Imprimitive non-powerful sign pattern matrices with maximum base

You et al. [L.H. You, J.Y. Shao, and H.Y. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007), pp. 285–300], obtained an upper bound of the bases for imprimitive non-powerful sign pattern matrices. In this article, we characterize those imprimitive non-powerful sign pattern matrices whose bases reach this upper bound.     

Construction and nonnegativity of sign idempotent sign pattern matrices

In this paper, we modify Eschenbach’s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them. We find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix. Additionally, if for a sign idempotent sign pattern matrix A there exists a signature matrix S such that SAS is nonnegative, we prove such S is unique up to multiplication by -1 if the signed digraph D(A) is not disconnected.     

Primitive zero-symmetric sign pattern matrices with the maximum base

In [B. Cheng, B. Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715-731], Cheng and Liu studied the bases of primitive zero-symmetric sign pattern matrices. The sharp upper bound of the bases was obtained. In this paper, we characterize the sign pattern matrices with the sharp bound.     

Primitive non-powerful sign pattern matrices with base 2

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, M ^k ?=?J _#. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M)?=?2, the minimum number of non-zero entries of M is 5n???8 or 5n???7 depending on whether n is even or odd.     

Coefficients of ergodicity and the scrambling index

For a primitive stochastic matrix S, upper bounds on the second largest modulus of an eigenvalue of S are very important, because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix. In this paper, we introduce the definition of the scrambling index for a primitive digraph. The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). We investigate the scrambling index for primitive digraphs, and give an upper bound on the scrambling index of a primitive digraph in terms of the order and the girth of the digraph. By doing so we provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index.     

Matrices with maximum kth local exponent in the class of doubly symmetric primitive matrices

Let A be a primitive matrix of order n, and let k be an integer with 1?k?n. The kth local exponent of A, is the smallest power of A for which there are k rows with no zero entry. We have recently obtained the maximum value for the kth local exponent of doubly symmetric primitive matrices of order n with 1?k?n. In this paper, we use the graph theoretical method to give a complete characterization of those doubly symmetric primitive matrices whose kth local exponent actually attain the maximum value.     

The period and base of a reducible sign pattern matrix

A square sign pattern matrix A (whose entries are ) is said to be powerful if all the powers A,A²,A³,…, are unambiguously defined. For a powerful pattern A, if A^l=A^l+p with l and p minimal, then l is called the base of A and p is called the period of Li et al. [On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101-120] characterized irreducible powerful sign pattern matrices. In this paper, we characterize reducible, powerful sign pattern matrices and give some new results on the period and base of a powerful sign pattern matrix.     

A relationship between subpermanents and the arithmetic-geometric mean inequality

Using the arithmetic-geometric mean inequality, we give bounds for k-subpermanents of nonnegative n×n matrices F. In the case k=n, we exhibit an n²-set S whose arithmetic and geometric means constitute upper and lower bounds for per(F)/n!. We offer sharpened versions of these bounds when F has zero-valued entries.     

本原不可幂带号有向图的lewin数的界

如果存在正整数k使得对于D中任意两点u和v(允许u=v),在D中都有从u到v的长为k的有向途径,则称有向图D是本原的.给有向图的每条弧赋以符号+1或者-1得到的图S称为带号有向图.如果带号有向图S中包含SSSD途径对,即包含两条有相同的起点,相同的终点,相同的长度,并且有不同的符号的途径对,则称S是不可幂的.在本文中,我们将Lewin M提出的lewin数的概念从本原有向图推广到本原不可幂带号有向图,给出了本原不可幂带号有向图S的lewin数l(S)的若干上界,并提出了一个公开问题.     

Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.     

本原对称无环带号有向图的基指数与基指数集

In this paper,we study the bases and base sets of primitive symmetric loop-free (generalized)signed digraphs on n vertices.We obtain sharp upper bounds of the bases,and show that the base sets of the classes of such digraphs are{2,3,...,2n-1}.We also give a new proof of an important result obtained by Cheng and Liu.     

Spaces of real matrices of fixed small rank

In this paper we investigate the maximal dimension for k-spaces of real matrices for small values of k. We also give some bounds for the dimension of spaces of real matrices of high rank.