Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/63847
Title: Spectral properties of odd-bipartite Z-tensors and their absolute tensors
Authors: Chen, H
Qi, LQ 
Keywords: H-Eigenvalue
Z-tensor
Odd-bipartite tensor
Absolute tensor
Issue Date: 2016
Publisher: Higher Education Press
Source: Frontiers of mathematics in China, 2016, v. 11, no. 3, p. 539-556 How to cite?
Journal: Frontiers of mathematics in China 
Abstract: Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Z-tensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.
URI: http://hdl.handle.net/10397/63847
ISSN: 1673-3452
EISSN: 1673-3576
DOI: 10.1007/s11464-016-0520-4
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