Sharp bounds for spectral radius of nonnegative weakly irreducible tensors

We obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. By using the technique of the representation associate matrix of a tensor and the associate directed graph of the matrix, the equality cases of the bounds are completely characterized by graph theory methods. Applying these bounds to a nonnegative irreducible matrix or a connected graph (digraph), we can improve the results of L. H. You, Y. J. Shu, and P. Z. Yuan [Linear Multilinear Algebra, 2017, 65(1): 113–128], and obtain some new or known results. Applying these bounds to a uniform hypergraph, we obtain some new results and improve some known results of X. Y. Yuan, M. Zhang, and M. Lu [Linear Algebra Appl., 2015, 484: 540–549]. Finally, we give a characterization of a strongly connected k-uniform directed hypergraph, and obtain some new results by applying these bounds to a uniform directed hypergraph.     

Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Brauer-type bounds for Hadamard product of nonnegative tensors

In this paper, we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set, which are shown to be sharper than the existing bounds established in the literature. The validity of the obtained results is theoretically and numerically tested.     

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

For a compact subset of symmetric with respect to conjugation and a continuous function, we obtain sharp conditions on and that insure that can be approximated uniformly on by polynomials with nonnegative coefficients. For a real Banach space, a closed but not necessarily normal cone with , and a bounded linear operator with , we use these approximation theorems to investigate when the spectral radius of belongs to its spectrum . A special case of our results is that if is a Hilbert space, is normal and the 1-dimensional Lebesgue measure of is zero, then . However, we also give an example of a normal operator (where is unitary and ) for which and .

     

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.     

实对称张量正定性的判定

实对称张量的正定性在自动控制系统稳定性、多项式全局优化、医疗影像降噪等问题中具有重要的应用价值.通过构造不同的正对角阵和运用不等式的放缩技巧,给出了H-张量新的判别条件.作为应用,给出了偶数阶实对称张量,即偶次齐次多项式正定性的新实用判定方法.相应数值算例表明了结果的有效性.     

On the Laplacian spectral radii of trees

Let Δ(T) and μ(T) denote the maximum degree and the Laplacian spectral radius of a tree T, respectively. Let T_n be the set of trees on n vertices, and . In this paper, we determine the two trees which take the first two largest values of μ(T) of the trees T in when . And among the trees in , the tree which alone minimizes the Laplacian spectral radius is characterized. We also prove that for two trees T₁ and T₂ in , if Δ(T₁)>Δ(T₂) and , then μ(T₁)>μ(T₂). As an application of these results, we give a general approach about extending the known ordering of trees in T_n by their Laplacian spectral radii.     

On the Laplacian spectral radii of bicyclic graphs

A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n+1. Let B(n) be the set of all bicyclic graphs on n vertices. In this paper, we obtain the first four largest Laplacian spectral radii among all the graphs in the class B(n) (n≥7) together with the corresponding graphs.     

On the perturbation of rank‐one symmetric tensors

The problem of symmetric rank‐one approximation of symmetric tensors is important in independent components analysis, also known as blind source separation, as well as polynomial optimization. We derive several perturbative results that are relevant to the well‐posedness of recovering rank‐one structure from approximately‐rank‐one symmetric tensors. We also specialize the analysis of the shifted symmetric higher‐order power method, an algorithm for computing symmetric tensor eigenvectors, to approximately‐rank‐one symmetric tensors. Copyright © 2012 John Wiley & Sons, Ltd.     

Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices.     

Laplacian and signless Laplacian spectral radii of graphs with fixed domination number

We determine the maximal Laplacian and signless Laplacian spectral radii for graphs with fixed number of vertices and domination number, and characterize the extremal graphs.     

A formula for the joint local spectral radius

We give a formula for the joint local spectral radius of a bounded subset of bounded linear operators on a Banach space in terms of the dual of .

     

Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor

An iterative method for finding the largest eigenvalue of a nonnegative tensor was proposed by Ng, Qi, and Zhou in 2009. In this paper, we establish an explicit linear convergence rate of the Ng–Qi–Zhou method for essentially positive tensors. Numerical results are given to demonstrate linear convergence of the Ng–Qi–Zhou algorithm for essentially positive tensors. Copyright © 2011 John Wiley & Sons, Ltd.     

On sharp bounds for spectral radius of nonnegative matrices

We give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with positive row sums using average 3-row sums, compare these bounds with the existing bounds using the average 2-row sums by examples, and apply them to the adjacency matrix and the signless Laplacian matrix of a digraph or a graph.     

判定亚正定矩阵的一个充要条件

给出了一个更一般的判定亚正定矩阵的充要条件.     

非负矩阵谱半径的一个新界值

在Wielandt定理的基础上进行了推广,得到了一种估计非负矩阵谱半径的新方法,数值例子显示了新方法所得到的结果更为精确.     

A constructive arbitrary‐degree Kronecker product decomposition of tensors

We generalize the matrix Kronecker product to tensors and propose the tensor Kronecker product singular value decomposition that decomposes a real k‐way tensor into a linear combination of tensor Kronecker products with an arbitrary number of d factors. We show how to construct , where each factor is also a k‐way tensor, thus including matrices (k=2) as a special case. This problem is readily solved by reshaping and permuting into a d‐way tensor, followed by a orthogonal polyadic decomposition. Moreover, we introduce the new notion of general symmetric tensors (encompassing symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors, etc.) and prove that when is structured then its factors will also inherit this structure.