H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph

The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.

m, argest adjacency, signless Laplacian, and Laplacian H-eigenvalues of loose paths

We investigate k-uniform loose paths. We show that the largest H- eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥ 3, we show that the largest H-eigenvalue of its adjacency tensor is （（1 ＋ √-5）/2）2/k when = 3 and ）λ（A） = 31/k when g = 4, respectively. For the case of l ≥ 5, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval （2, 3） when l≥ 5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.

On computing minimal H-eigenvalue of sign-structured tensors

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.

Least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices

Let G be a connected hypergraph with even uniformity,which contains cut vertices.Then G is the coalescence of two nontrivial connected sub-hypergraphs(called branches)at a cut vertex.Let A(G)be the adjacency tensor of G.The least H-eigenvalue of A(G)refers to the least real eigenvalue of A(G)associated with a real eigenvector.In this paper,we obtain a perturbation result on the least H-eigenvalue of A(G)when a branch of G attached at one vertex is relocated to another vertex,and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.

Largest H-eigenvalue of uniform s-hypertrees

The k-uniform s-hypertree G = （V, E） is an s-hypergraph, where 1 ≤ s ≤ k - 1, and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree A. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just （△S/k）.

Column sufficient tensors and tensor complementarity problems

Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.

The Adjacency and Signless Laplacian Spectra of Cored Hypergraphs and Power Hypergraphs

In this paper,we study the adjacency and signless Laplacian tensors of cored hypergraphs and power hypergraphs.We investigate the properties of their adjacency and signless Laplacian H-eigenvalues.Especially,we find out the largest H-eigenvalues of adjacency and signless Laplacian tensors for uniform squids.We also compute the H-spectra of sunflowers and some numerical results are reported for the H-spectra.

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

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.

High-order sum-of-squares structured tensors:theory and applications

Tensor decomposition is an important research area with numerous applications in data mining and computational neuroscience.An important class of tensor decomposition is sum-of-squares(SOS)tensor decomposition.SOS tensor decomposition has a close connection with SOS polynomials,and SOS polynomials are very important in polynomial theory and polynomial optimization.In this paper,we give a detailed survey on recent advances of high-order SOS tensors and their applications.It first shows that several classes of symmetric structured tensors available in the literature have SOS decomposition in the even order symmetric case.Then,the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones are established.Further,a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent is provided,and the exact SOS-width for the cone consists of all such tensors with SOS decomposition is identified.Some potential research directions in the future are also listed in this paper.