Shenglong Hu

Algebraic connectivity of an even uniform hypergraph

We generalize Laplacian matrices for graphs to Laplacian tensors for even uniform hypergraphs and set some foundations for the spectral hypergraph theory based upon Laplacian tensors. Especially, algebraic connectivity of an even uniform hypergraph based on Z-eigenvalues of the corresponding Laplacian tensor is introduced and its connections with edge connectivity and vertex connectivity are discussed.

On determinants and eigenvalue theory of tensors

We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard’s inequality, Geršgrin’s inequality and Minikowski’s inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.

Strictly nonnegative tensors and nonnegative tensor partition

In this paper, we introduce a new class of nonnegative tensors — strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T , there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

The eigenvectors associated with the zero eigenvalues of the Laplacian and signless Laplacian tensors of a uniform hypergraph

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a k-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a canonical regularization, the phases of the components of these eigenvectors only can take some uniformly distributed values in {exp( k ) | j ∈ [k]}. These eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors with minimal support is called minimal. The minimal canonical H-eigenvectors characterize the even (odd)-bipartite connected components of the hypergraph and vice versa, and the minimal canonical N-eigenvectors characterize some multi-partite connected components of the hypergraph and vice versa.

The Laplacian of a uniform hypergraph

Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

k

A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems

-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval