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

作　　者：Haibin CHEN Liqun QI 

机构地区：[1]Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong,China

出　　处：《Frontiers of Mathematics in China》2016年第3期539-556,共18页中国高等学校学术文摘·数学（英文）

基　　金：Acknowledgements The authors were thankful to the anonymous relerees for the meaningful suggestions to improve the paper. The first author＇s work was supported in part by the National Natural Science Foundation of China （Grant No. 11171180） and the second author＇s work was supported by the Hong Kong Research Grant Council （Grant Nos. PolyU 502111, 501212, 501913, 15302114）.

摘　　要：Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite （weakly odd-bipartie） and even-bipartite （weakly even- bipartite） 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.Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite （weakly odd-bipartie） and even-bipartite （weakly even- bipartite） 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.

关 键 词：H-Eigenvalue Z-tensor odd-bipartite tensor absolute tensor 

分 类 号：O172.2[理学—数学] TB301[理学—基础数学]