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H+EIGENVALUES OF LAPLACIAN AND SIGNLESS LAPLACIAN TENSORS
, 2014
"... We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H+eigenvalues, i.e., Heigenvalues with nonnegative Heigenvectors, and H++eigenvalues, i.e., Heigenvalues with positive Heigenvectors. We show that each of ..."
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Cited by 29 (19 self)
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We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H+eigenvalues, i.e., Heigenvalues with nonnegative Heigenvectors, and H++eigenvalues, i.e., Heigenvalues with positive Heigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one H++eigenvalue, but has several other H+eigenvalues. We identify their largest and smallest H+eigenvalues, and establish some maximum and minimum properties of these H+eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.
Symmetric nonnegative tensors and copositive tensors
 LINEAR ALGEBRA AND ITS APPLICATIONS 439 (2013) 228–238
, 2013
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A survey on the spectral theory of nonnegative tensors
 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
, 2013
"... This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications. After a brief review of the basic definitions on tensors, the Heigenvalue problem and the Zeigenvalue problem for tensors are studied separately. To the Heigenvalue problem for non ..."
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Cited by 10 (6 self)
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This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications. After a brief review of the basic definitions on tensors, the Heigenvalue problem and the Zeigenvalue problem for tensors are studied separately. To the Heigenvalue problem for nonnegative tensors, the whole Perron–Frobenius theory for nonnegative matrices is completely extended, while to the Zeigenvalue problem, there are many distinctions and are studied carefully in details. Numerical methods are also discussed. Three kinds of applications are studied: higher order Markov chains, spectral theory of
Finding the maximum eigenvalue of essentially nonnegative symmetric tensors via sum of squares programming
, 2012
"... Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is nonnegative in the sense that all of its entries are nonnegative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on ..."
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Cited by 8 (8 self)
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Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is nonnegative in the sense that all of its entries are nonnegative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on a PerronFrobenius type theorem for nonnegative tensors. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in turn, can be equivalently rewritten as a semidefinite programming problem. Then, using this sum of squares programming problem, we also provide upper as well as lower estimate for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor.
The dominant eigenvalue of an essentially nonnegative tensor
, 2013
"... It is well known that the dominant eigenvalue of a real essentially nonnegative matrix is a convex function of its diagonal entries. This convexity is of practical importance in population biology, graph theory, demography, analytic hierarchy process, and so on. In this paper, the concept of essenti ..."
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Cited by 5 (5 self)
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It is well known that the dominant eigenvalue of a real essentially nonnegative matrix is a convex function of its diagonal entries. This convexity is of practical importance in population biology, graph theory, demography, analytic hierarchy process, and so on. In this paper, the concept of essentially nonnegativity is extended from matrices to higherorder tensors, and the convexity and log convexity of dominant eigenvalues for such a class of tensors are established. Particularly, for any nonnegative tensor, the spectral radius turns out to be the dominant eigenvalue and hence possesses these convexities. Finally, an algorithm is given to calculate the dominant eigenvalue, and numerical results are reported to show the effectiveness of the
EIGENVALUE ANALYSIS OF CONSTRAINED MINIMIZATION PROBLEM FOR HOMOGENEOUS POLYNOMIAL
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, 2014
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