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1.

Inertially arbitrary patterns

**总被引：11，自引：0，他引：11**An

*n*×*n*sign pattern matrix*A*is an inertially arbitrary pattern (**IAP**) if each non-negative triple (*r**s**t*) with*r*+*s*+*t*=*n*is the inertia of a matrix with sign pattern*A*. This paper considers the*n*×*n*(*n*≥2) skew-symmetric sign pattern*S*with each upper off-diagonal entry positive, the (1,1) entry negative, the (_{n}*n**n*) entry positive, and every other diagonal entry zero. We prove that*S*is an_{n}**IAP**. 相似文献2.

Exponents of 2-coloring of symmetric digraphs

**总被引：1，自引：0，他引：1**A 2-coloring (

*G*_{1},*G*_{2}) of a digraph is 2-primitive if there exist nonnegative integers*h*and*k*with*h*+*k*>0 such that for each ordered pair (*u*,*v*) of vertices there exists an (*h*,*k*)-walk in (*G*_{1},*G*_{2}) from*u*to*v*. The exponent of (*G*_{1},*G*_{2}) is the minimum value of*h*+*k*taken over all such*h*and*k*. In this paper, we consider 2-colorings of strongly connected symmetric digraphs with loops, establish necessary and sufficient conditions for these to be 2-primitive and determine an upper bound on their exponents. We also characterize the 2-colored digraphs that attain the upper bound and the exponent set for this family of digraphs on*n*vertices. 相似文献3.

A ± sign pattern is a matrix whose entries are in the set {+,–}. An

*n*×*n*± sign pattern*A*allows orthogonality if there exists a real orthogonal matrix*B*in the qualitative class of*A*. In this paper, we prove that for*n*3 there is an*n*×*n*± sign pattern*A*allowing orthogonality with exactly*k*negative entries if and only if*n*–1*k**n*^{2}–*n*+1.Research supported by Shanxi Natural Science Foundation 20011006, 20041010Final version received: October 22, 2003 相似文献4.

A sign pattern

*A*is a ± sign pattern if*A*has no zero entries.*A*allows orthogonality if there exists a real orthogonal matrix*B*whose sign pattern equals*A*. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with*n*− 1 ⩽ N_{−}(*A*) ⩽*n*+ 1 to allow orthogonality. 相似文献5.

The inertia set of a symmetric sign pattern

*A*is the set*i*(*A*) = {*i*(*B*) |*B*=*B*^{T}∈*Q*(*A*)}, where*i*(*B*) denotes the inertia of real symmetric matrix*B*, and*Q*(*A*) denotes the sign pattern class of*A*. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern*A*in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns*A*with zero diagonal that require unique inertia. 相似文献6.

The scrambling index of an

*n*×*n*primitive Boolean matrix*A*is the smallest positive integer*k*such that*A*^{ k }(*A*^{T})^{ k }=*J*, where*A*^{T}denotes the transpose of*A*and*J*denotes the*n*×*n*all ones matrix. For an*m*×*n*Boolean matrix*M*, its Boolean rank*b*(*M*) is the smallest positive integer*b*such that*M*=*AB*for some*m*×*b*Boolean matrix*A*and*b*×*n*Boolean matrix*B*. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an*n*×*n*primitive matrix*M*in terms of its Boolean rank*b*(*M*), and they also characterized all primitive matrices that achieve the upper bound. In this paper, we characterize primitive Boolean matrices that achieve the second largest scrambling index in terms of their Boolean rank. 相似文献7.

An

*n*×*n*ray pattern matrix*S*is said to be spectrally arbitrary if for every monic*n*th degree polynomial*f*(*λ*) with coefficients from*C*, there is a complex matrix in the ray pattern class of*S*such that its characteristic polynomial is*f*(*λ*). In this article we give new classes of spectrally arbitrary ray pattern matrices. 相似文献8.

In this paper, we study two classes of primitive digraphs, and show that there are

*k*-colorings that are*k*-primitive. 相似文献9.

An

*n*×*n*sign pattern*A*is said to be potentially nilpotent if there exists a nilpotent real matrix*B*with the same sign pattern as*A*. Let*D*_{n,r}be an*n*×*n*sign pattern with 2 ≤*r*≤*n*such that the superdiagonal and the (*n*,*n*) entries are positive, the (*i*, 1) (*i*= 1,...,*r*) and (*i*,*i*−*r*+ 1) (*i*=*r*+ 1,...,*n*) entries are negative, and zeros elsewhere. We prove that for*r*≥ 3 and*n*≥ 4*r*− 2, the sign pattern*D*_{n,r}is not potentially nilpotent, and so not spectrally arbitrary. 相似文献10.

Wei Fang Wei Gao Yubin Gao Fei Gong Guangming Jing Zhongshan Li Yanling Shao Lihua Zhang 《Czechoslovak Mathematical Journal》2016,66(3):895-911

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,−, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix

*A*is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of*A*. Using a correspondence between sign patterns with minimum rank*r*≥ 2 and point-hyperplane configurations in R^{r−1}and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every*d*-polytope determines a nonnegative sign pattern with minimum rank*d*+ 1 that has a (*d*+ 1) × (*d*+ 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3*m*, 3*n*} zero entries in any condensed nonnegative*m*×*n*sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established. 相似文献