The kth lower bases of primitive non-powerful signed digraphs

In [J. Shao, L. You, H. Shan, Bound on the bases of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 285-300], the authors extended the concept of the base from powerful sign pattern matrices to non-powerful irreducible sign pattern matrices. Recently, the kth local bases and the kth upper bases, which are generalizations of the bases, of primitive non-powerful signed digraphs were introduced. In this paper, we introduce a new parameter called the kth lower bases of primitive non-powerful signed digraphs and obtain some bounds for it. For some cases, the bounds we obtain are best possible and the extremal signed digraphs are characterized, respectively. Moreover, we show that there exist “gaps” in the kth lower bases set of primitive non-powerful signed digraphs.     

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.     

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.     

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.     

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.     

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.     

The base sets of quasi-primitive zero-symmetric sign pattern matrices with zero trace

Cheng and Liu [Bo Cheng, Bolian Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715-731] showed that the base set of quasi-primitive zero-symmetric (generalized) sign pattern matrices is {1,2,…,2n}. The matrices with zero trace play a prominent role in matrix theory. In this paper, we investigate the bases of quasi-primitive zero-symmetric (generalized) sign pattern matrices with zero trace and prove that the base set of such matrices is {2,3,…,2n-1}.     

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.     

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.     

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.     

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.     

3阶不可约符号模式的精细惯量极小临界集

Let S be a nonempty, proper subset of all possible refined inertias of real matrices of order n. The set S is a critical set of refined inertias for irreducible sign patterns of order n,if for each n × n irreducible sign pattern A, the condition S ? ri(A) is sufficient for A to be refined inertially arbitrary. If no proper subset of S is a critical set of refined inertias, then S is a minimal critical set of refined inertias for irreducible sign patterns of order n.All minimal critical sets of refined inertias for full sign patterns of order 3 have been identified in [Wei GAO, Zhongshan LI, Lihua ZHANG, The minimal critical sets of refined inertias for 3×3 full sign patterns, Linear Algebra Appl. 458(2014), 183–196]. In this paper, the minimal critical sets of refined inertias for irreducible sign patterns of order 3 are identified.     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

The proof on the conjecture of extremal graphs for the kth eigenvalues of trees

We consider the only remaining unsolved case n≡0 (mod k) for the largest kth eigenvalue λ_k.of trees with n vertices. In this paper, the conjecture for this problem in [Shao Jia-yu, On the largest kth eignevalues of trees, Linear Algebra Appl. 221 (1995) 131] is proved and (from this) the complete solution to this problem, the best upper bound and the extremal trees of λ_k, is given in general cases above.     

A structural characterization of real k-potent matrices

Some well-known characterizations of nonnegative k-potent matrices have been obtained by Flor [P. Flor, On groups of nonnegative matrices, Compositio Math. 21 (1969), pp. 376–382.] and Jeter and Pye [M. Jeter and W. Pye, Nonnegative (s,?t)-potent matrices, Linear Algebra Appl. 45 (1982), pp. 109–121.]. In this article, we obtain a structural characterization of a real k-potent matrix A, provided that (sgn(A)) ^k+1 is unambiguously defined, regardless of whether A is nonnegative or not.     

具有最小弧数且基指数为3的本原不可幂对称无环带号有向图

Let S be a primitive non-powerful symmetric loop-free signed digraph on even n vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434(5), 1215-1227], authors conjectured that D is the underlying digraph of S with exp(D) = 3 if and only if D is isomorphic to ED n,3,3 , where ED n,3,3 = (V, A) is a digraph with V = {1, 2, . . . , n}, A = {(1, i), (i, 1) | 3≤i≤n} ∪ {(2i-1, 2i), (2i, 2i-1) | 2≤i≤ n/2 } ∪ {(2, 3), (3, 2), (2, 4), (4, 2)}). In this paper, we show the conjecture is true and completely characterize the underlying digraphs which have base 3 and the minimum number of arcs.     

Cocharacters of Bilinear Mappings and Graded Matrices

Let M _k (F) be the algebra of k ×k matrices over a field F of characteristic 0. If G is any group, we endow M _k (F) with the elementary grading induced by the k-tuple (1,...,1,g) where g?∈?G, g ²?≠?1. Then the graded identities of M _k (F) depending only on variables of homogeneous degree g and g ^???1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g ^???1 and we find an explicit decomposition of the corresponding graded cocharacter into irreducibles.     

A note on the kth eigenvalue of trees

In [Linear Algebra Appl. 149 (1991) 19-34], Shao proved that for a tree T on n vertices, the kth eigenvalue

     

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 index of maximum ambiguous density for irreducible non-powerful sign pattern matrices

Let INS_n,p be the set of n×n irreducible non-powerful (generalized) sign pattern matrices with period p, and let AINS_n,p. In this paper, we introduce a new parameter called the index of maximum ambiguous density of A. Furthermore, the generalized index of maximum ambiguous density of A, which generalizes the concept of the index of maximum ambiguous density, is introduced. Moreover, some bounds on these indices are obtained, and we exhibit a system of gaps in the set of the index of maximum ambiguous density for AINS_n,p. Finally, the index and the generalized index of maximum ambiguous density for irreducible non-powerful zero-symmetric sign pattern matrices are discussed.