奇偶$Z$ -张量及其绝对张量的谱性质
Some Spectral Properties of Odd-Bipartite $Z$-Tensors and Their Absolute Tensors
摘要：
受奇二部超图和偶二部超图的启发，我们定义了奇二部（弱奇二部）张量和偶二部（弱偶二部）张量。证明了所有偶数阶奇二分张量都是不可约张量，而所有偶数阶二分张量都是可约张量，无论其阶数的奇偶性如何。基于奇偶张量的性质，研究了具有非负对角元的$Z$ -张量的最大H-特征值与该$Z$ -张量的绝对张量的最大H-特征值之间的关系.当阶为偶数且$Z$ -张量弱不可约时，证明了$Z$ -张量的最大H-特征值与该$Z$ -张量的绝对张量的最大H-特征值相等，当且仅当$Z$ -张量弱奇二分.实例验证了结论的真实性。证明了对角元非负的对称$Z$ -张量与$Z$ -张量的绝对张量对角相似的充要条件是$Z$ -张量为偶数阶且弱奇二分.然后证明了当偶数阶对称的具有非负对角元的$Z$ -张量是弱不可约的时，$Z$ -张量的谱与该$Z$ -张量的绝对张量的谱的相等性可以用它们的谱半径相等来刻画.
英文摘要：
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.
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