Spectral properties of odd-bipartite <i>Z</i>-tensors and their absolute tensors

doi:10.1007/s11464-016-0520-4

Front. Math. China

2016, Vol. 11

Issue (3) : 539-556 https://doi.org/10.1007/s11464-016-0520-4

RESEARCH ARTICLE

Spectral properties of odd-bipartite Z-tensors and their absolute tensors

Haibin CHEN(

),Liqun QI

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

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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.

Keywords H-Eigenvalue Z-tensor odd-bipartite tensor absolute tensor

Corresponding Author(s): Haibin CHEN

Issue Date: 17 May 2016

Cite this article:

Haibin CHEN,Liqun QI. Spectral properties of odd-bipartite Z-tensors and their absolute tensors[J]. Front. Math. China, 2016, 11(3): 539-556.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0520-4
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I3/539

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