Some Spectral Properties of Odd-Bipartite -Tensors and Their Absolute Tensors
arXiv:1503.07959
Abstract
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 -tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that -tensor. When the order is even and the -tensor is weakly irreducible, we prove that the largest H-eigenvalue of the -tensor and the largest H-eigenvalue of the absolute tensor of that -tensor are equal, if and only if the -tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric -tensor with nonnegative diagonal entries and the absolute tensor of the -tensor are diagonal similar, if and only if the -tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric -tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the -tensor and the spectrum of absolute tensor of that -tensor, can be characterized by the equality of their spectral radii.