Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with the least H-eigenvalue. If $\mathcal{A}$ is further combinatorial symmetric, the number of such eigenvectors can be obtained explicitly by the Smith normal form of the incidence matrix of $\mathcal{A}$. When applying to a connected uniform hypergraph $G$, we prove that the number of Laplacian eigenvectors of $G$ associated with the zero eigenvalue is equal to the the number of adjacency eigenvectors of $G$ associated with the spectral radius, which is also equal to the number of signless Laplacian eigenvectors of $G$ associated with the zero eigenvalue if zero is an signless Laplacian eigenvalue

The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

Let $\mathcal{A}$ be a weakly irreducible nonnegative tensor with spectral radius $\rho(\mathcal{A})$. Let $\mathfrak{D}$ (respectively, $\mathfrak{D}^{(0)}$) be the set of normalized diagonal matrices arising from the eigenvectors of $\mathcal{A}$ corresponding to the eigenvalues with modulus $\rho(\mathcal{A})$ (respectively, the eigenvalue $\rho(\mathcal{A})$). It is shown that $\mathfrak{D}$ is an abelian group containing $\mathfrak{D}^{(0)}$ as a subgroup, which acts transitively on the set $\{e^{\mathbf{i} \frac{2 \pi j}{\ell}}\mathcal{A}:j =0,1, \ldots,\ell-1\}$, where $|\mathfrak{D}/\mathfrak{D}^{(0)}|=\ell$ and $\mathfrak{D}^{(0)}$ is the stabilizer of $\mathcal{A}$. The spectral symmetry of $\mathcal{A}$ is characterized by the group $\mathfrak{D}/\mathfrak{D}^{(0)}$, and $\mathcal{A}$ is called spectral $\ell$-symmetric. We obtain the structural information of $\mathcal{A}$ by analyzing the property of $\mathfrak{D}$, especially for connected hypergraphs we get some results on the edge distribution and coloring. If moreover $\mathcal{A}$ is symmetric, we prove that $\mathcal{A}$ is spectral $\ell$-symmetric if and only if it is $(m,\ell)$-colorable. We characterize the spectral $\ell$-symmetry of a tensor by using its generalized traces, and show that for an arbitrarily given integer $m \ge 3$ and each positive integer $\ell$ with $\ell \mid m$, there always exists an $m$-uniform hypergraph $G$ such that $G$ is spectral $\ell$-symmetric

Eigenvectors of Laplacian or signless Laplacian of Hypergraphs Associated with Zero Eigenvalue

Let $G$ be a connected $m$-uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of $G$ associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of $G$. By means of the incidence matrix of $G$, the number of first Laplacian or signless Laplaican (H- or N-)eigenvectors can be get explicitly by solving the Smith normal form of the incidence matrix over $\mathbb{Z}_m$ or $\mathbb{Z}_2$. Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signless Laplacian (H-)eigenvectors when zero is an (H-)eigenvalue of the signless Laplacian tensor. We establish a connection between first Laplacian (signless Laplacian) H-eigenvectors and the even (odd) bipartitions of $G$

Adjacency Spectra of Random and Uniform Hypergraphs

We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all-ones hypermatrix. Several of the ingredients along a possible path to this conjecture are established, and may be of independent interest in spectral hypergraph/hypermatrix theory. In particular, we provide a bound on the spectral radius of the symmetric Bernoulli hyperensemble, and show that the spectrum of the complete $k$-uniform hypergraph for $k=2,3$ is close to that of an appropriately scaled all-ones hypermatrix.Comment: 22 pages, no figure

The HH-spectrum of a generalized power hypergraph

The generalized power of a simple graph $G$, denoted by $G^{k,s}$, is obtained from $G$ by blowing up each vertex into an $s$-set and each edge into a $k$-set, where $1 \le s \le \frac{k}{2}$. When $s < \frac{k}{2}$, $G^{k,s}$ is always odd-bipartite. It is known that $G^{k,{k \over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and $G^{k,{k \over 2}}$ has the same adjacency (respectively, signless Laplacian) spectral radius as $G$. In this paper, we prove that, regardless of multiplicities, the $H$-spectrum of \A(G^{k,\frac{k}{2}}) (respectively, \Q(G^{k,\frac{k}{2}})) consists of all eigenvalues of the adjacency matrices (respectively, the signless Laplacian matrices) of the connected induced subgraphs (respectively, modified induced subgraphs) of $G$. As a corollary, $G^{k,{k \over 2}}$ has the same least adjacency (respectively, least signless Laplacian) $H$-eigenvalue as $G$. We also discuss the limit points of the least adjacency $H$-eigenvalues of hypergraphs, and construct a sequence of non-odd-bipartite hypergraphs whose least adjacency $H$-eigenvalues converge to $-\sqrt{2+\sqrt{5}}$.Comment: arXiv admin note: text overlap with arXiv:1408.330

The largest HH-eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs

Let $G$ be a simple graph or hypergraph, and let $A(G),L(G),Q(G)$ be the adjacency, Laplacian and signless Laplacian tensors of $G$ respectively. The largest $H$-eigenvalues (resp., the spectral radii) of $L(G),Q(G)$ are denoted respectively by $\lambda_{\max}^L(G), \lambda_{\max}^Q(G)$ (resp., $\rho^L(G), \rho^Q(G)$). For a connected non-bipartite simple graph $G$, $\lambda_{\max}^L(G)=\rho^L(G) < \rho^Q(G)$. But this does not hold for non-odd-bipartite hypergraphs. We will investigate this problem by considering a class of generalized power hypergraphs $G^{k,\frac{k}{2}}$, which are constructed from simple connected graphs $G$ by blowing up each vertex of $G$ into a $\frac{k}{2}$-set and preserving the adjacency of vertices. Suppose that $G$ is non-bipartite, or equivalently $G^{k,\frac{k}{2}}$ is non-odd-bipartite. We get the following spectral properties: (1) $\rho^L(G^{k,{k \over 2}}) =\rho^Q(G^{k,{k \over 2}})$ if and only if $k$ is a multiple of $4$; in this case $\lambda_{\max}^L(G^{k,\frac{k}{2}})<\rho^L(G^{k,{k \over 2}})$. (2) If $k\equiv 2 (\!\!\!\mod 4)$, then for sufficiently large $k$, $\lambda_{\max}^L(G^{k,\frac{k}{2}})<\rho^L(G^{k,{k \over 2}})$. Motivated by the study of hypergraphs $G^{k,\frac{k}{2}}$, for a connected non-odd-bipartite hypergraph $G$, we give a characterization of $L(G)$ and $Q(G)$ having the same spectra or the spectrum of $A(G)$ being symmetric with respect to the origin, that is, $L(G)$ and $Q(G)$, or $A(G)$ and $-A(G)$ are similar via a complex (necessarily non-real) diagonal matrix with modular-$1$ diagonal entries. So we give an answer to a question raised by Shao et al., that is, for a non-odd-bipartite hypergraph $G$, that $L(G)$ and $Q(G)$ have the same spectra can not imply they have the same $H$-spectra

Some Spectral Properties and Characterizations of Connected Odd-bipartite Uniform Hypergraphs

A $k$-uniform hypergraph $G=(V,E)$ is called odd-bipartite ([5]), if $k$ is even and there exists some proper subset $V_1$ of $V$ such that each edge of $G$ contains odd number of vertices in $V_1$. Odd-bipartite hypergraphs are generalizations of the ordinary bipartite graphs. We study the spectral properties of the connected odd-bipartite hypergraphs. We prove that the Laplacian H-spectrum and signless Laplacian H-spectrum of a connected $k$-uniform hypergraph $G$ are equal if and only if $k$ is even and $G$ is odd-bipartite. We further give several spectral characterizations of the connected odd-bipartite hypergraphs. We also give a characterization for a connected $k$-uniform hypergraph whose Laplacian spectral radius and signless Laplacian spectral radius are equal, thus provide an answer to a question raised in [9]. By showing that the Cartesian product $G\Box H$ of two odd-bipartite $k$-uniform hypergraphs is still odd-bipartite, we determine that the Laplacian spectral radius of $G\Box H$ is the sum of the Laplacian spectral radii of $G$ and $H$, when $G$ and $H$ are both connected odd-bipartite.Comment: 16 page

On the principal eigenvectors of uniform hypergraphs

Let $\mathcal{A}(H)$ be the adjacency tensor of $r$-uniform hypergraph $H$. If $H$ is connected, the unique positive eigenvector $x=(x_1,x_2,\ldots,x_n)^{\mathrm{T}}$ with $||x||_r=1$ corresponding to spectral radius $\rho(H)$ is called the principal eigenvector of $H$. The maximum and minimum entries of $x$ are denoted by $x_{\max}$ and $x_{\min}$, respectively. In this paper, we investigate the bounds of $x_{\max}$ and $x_{\min}$ in the principal eigenvector of $H$. Meanwhile, we also obtain some bounds of the ratio $x_i/x_j$ for $i$, $j\in [n]$ as well as the principal ratio $\gamma(H)=x_{\max}/x_{\min}$ of $H$. As an application of these results we finally give an estimate of the gap of spectral radii between $H$ and its proper sub-hypergraph $H'$.Comment: In this version, we corrected a reference for the fact Page 6 Line 1, which shoud be [15], not [5] as befor

Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process

Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B$_0$ tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is $m$th order stationary, where $m$ is even, then its $m$th order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for a circulant tensor to be positive semi-definite

Eigenvariety of Nonnegative Symmetric Weakly Irreducible Tensors Associated with Spectral Radius and Its Application to Hypergraphs

For a nonnegative symmetric weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, there may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. In this paper we proved that such projective eigenvariety admits a module structure, which is determined by the support of the tensor and can be characterized explicitly by solving the Smith normal form of the incidence matrix of the tensor. We introduced two parameters: the stabilizing index and the stabilizing dimension of the tensor, where the former is exactly the cardinality of the projective eigenvariety and the latter is the composition length of the projective eigenvariety as a module. We give some upper bounds for the two parameters, and characterize the case that there is only one eigenvector of the tensor corresponding to the spectral radius, i.e. the Perron vector. By applying the above results to the adjacency tensor of a connected uniform hypergraph, we give some upper bounds for the two parameters in terms of the structural parameters of the hypergraph such as path cover number, matching number and the maximum length of paths