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Yi-Zheng Fan

Publications and source records attributed to Yi-Zheng Fan.

At least 55 records · Page 3Linked to original sources
arXiv2019

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 $G$.

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arXiv2018

The cyclic index of adjacency tensor of generalized power hypergraphs

Let $G$ be a $t$-uniform hypergraph, and let $c(G)$ denote the cyclic index of the adjacency tensor of $G$. Let $m,s,t$ be positive integers such that $t \ge 2$, $s \ge 2$ and $m=st$. The generalized power $G^{m,s}$ of $G$ is obtained from $G$ by blowing up each vertex into an $s$-set and preserving the adjacency relation. It was conjectured that $c(G^{m,s})=s \cdot c(G)$. In this paper we show that the conjecture is false by giving a counterexample, and give some sufficient conditions for the conjecture holding. Finally we give an equivalent characterization of the equality in the conjecture by using a matrix equation over $\mathbb{Z}_m$.

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arXiv2018

The Dimension of Eigenvariety of Nonnegative Tensors Associated with Spectral Radius

For a nonnegative 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, it 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 prove that the dimension of the above projective eigenvariety is zero, i.e. there are finite many eigenvectors associated with the spectral radius up to a scalar. For a general nonnegative tensor, we characterize the nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs.

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arXiv2018

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.

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arXiv2017

The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

Let $\mathcal{A}$ be a weakly irreducible nonnegative tensor with spectral radius $ρ(\mathcal{A})$. 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 $ρ(\mathcal{A})$ (respectively, the eigenvalue $ρ(\mathcal{A})$). It is shown that $\mathfrak{D}$ is an abelian group containing $\mathfrak{D}^{(0)}$ as a subgroup, which acts transitively on the set $\{e^{\mathbf{i} \frac{2 πj}{\ell}}\mathcal{A}:j =0,1, \ldots,\ell-1\}$, 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.

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arXiv2015

The largest $H$-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 $λ_{\max}^L(G), λ_{\max}^Q(G)$ (resp., $ρ^L(G), ρ^Q(G)$). For a connected non-bipartite simple graph $G$, $λ_{\max}^L(G)=ρ^L(G) < ρ^Q(G)$. 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) $ρ^L(G^{k,{k \over 2}}) =ρ^Q(G^{k,{k \over 2}})$ if and only if $k$ is a multiple of $4$; in this case $λ_{\max}^L(G^{k,\frac{k}{2}})<ρ^L(G^{k,{k \over 2}})$. (2) If $k\equiv 2 (\!\!\!\mod 4)$, then for sufficiently large $k$, $λ_{\max}^L(G^{k,\frac{k}{2}})<ρ^L(G^{k,{k \over 2}})$. 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.

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arXiv2015

Maximizing spectral radii of uniform hypergraphs with few edges

In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic hypergraphs, linear or power unicyclic hypergraphs with given girth, linear or power bicyclic hypergraphs, respectively.

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arXiv2015

The $H$-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 $-\sqrt{2+\sqrt{5}}$.

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arXiv2015

On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs

In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph $G$, we construct a generalized power of $G$, denoted by $G^{k,s}$, which is obtained from $G$ by blowing up each vertex into a $k$-set and each edge into a $(k-2s)$-set, where $s \le k/2$. When $s < k/2$, $G^{k,s}$ is always odd-bipartite. We show that $G^{k,{k \over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and find that $G^{k,{k \over 2}}$ has the same adjacency (respectively, signless Laplacian) spectral radius as $G$. So the results involving the adjacency or signless Laplacian spectral radius of a simple graph $G$ hold for $G^{k,{k \over 2}}$. In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs $G^{k,{k \over 2}}$ of fixed order, and prove that $\sqrt{2+\sqrt{5}}$ is the smallest limit point of the non-odd-bipartite hypergraphs $G^{k,{k \over 2}}$. In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.

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arXiv2014

Bounds for the positive or negative inertia index of a graph

Let $G$ be a graph and let $A(G)$ be adjacency matrix of $G$.The positive inertia index (respectively, the negative inertia index) of $G$, denoted by $p(G)$ (respectively, $n(G)$), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of $A(G)$. In this paper, we present the bounds for $p(G)$ and $n(G)$ as follows: $$m(G)-c(G)\leq p(G)\leq m(G)+c(G), \ m(G)-c(G)\leq n(G)\leq m(G)+c(G),$$ where $m(G)$ and $c(G)$ are respectively the matching number and the cyclomatic number of $G$. Furthermore, we characterize the graphs which attain the upper bounds or the lower bounds respectively.

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arXiv2014

The spectral radius of the square of graphs

The square of a connected graph $G$ is obtained from $G$ by adding an edge between every pair of vertices at distance $2$. In this paper we give some upper or lower bounds for the spectral radius of the square of connected graphs, trees and unicyclic graphs respectively.We also investigate the spectral radius of the square of unicyclic graphs with given girth or trees with fixed diameter.

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arXiv2013

Graph Automorphisms from the Geometric Viewpoint

An automorphism of a graph $G=(V,E)$ is a bijective map $ϕ$ from $V$ to itself such that $ϕ(v_i)ϕ(v_j)\in E$ $\Leftrightarrow$ $v_i v_j\in E$ for any two vertices $v_i$ and $v_j$. Denote by $\mathfrak{G}$ the group consisting of all automorphisms of $G$. Apparently, an automorphism of $G$ can be regarded as a permutation on $[n]=\{1,\ldots,n\}$, provided that $G$ has $n$ vertices. For each permutation $σ$ on $[n]$, there is a natural action on any given vector $\boldsymbol{u}=(u_1,\ldots,u_n)^t\in \mathbb{C}^n$ such that $σ\boldsymbol{u}=(u_{σ^{-1}1},u_{σ^{-1}2},\ldots,u_{σ^{-1} n})^t$, so $σ$ can be viewed as a linear operator on $\mathbb{C}^n$. Accordingly, one can formulate a characterization to the automorphisms of $G$, {\it i.e.,} $σ$ is an automorphism of $G$ if and only if every eigenspace of $\mathbf{A}(G)$ is $σ$-invariant, where $\mathbf{A}(G)$ is the adjacency matrix of $G$. Consequently, every eigenspace of $\mathbf{A}(G)$ is $\mathfrak{G}$-invariant, which is equivalent to that for any eigenvector $\boldsymbol{v}$ of $\mathbf{A}(G)$ corresponding to the eigenvalue $λ$, $\mathrm{span}(\mathfrak{G}\boldsymbol{v})$ is a subspace of the eigenspace $V_λ$. By virtue of the linear representation of the automorphism group $\mathfrak{G}$, we characterize those extremal vectors $\boldsymbol{v}$ in an eigenspace of $\mathbf{A}(G)$ so that $\mathrm{dim}~\mathrm{span}(\mathfrak{G}\boldsymbol{v})$ can attain extremal values, and furthermore, we determine the exact value of $\mathrm{dim}~\mathrm{span}(\mathfrak{G}\boldsymbol{v})$ for any eigenvector $\boldsymbol{v}$ of $\mathbf{A}(G)$.

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arXiv2013

A lower bound for the algebraic connectivity of a graph in terms of the domination number

We investigate how the algebraic connectivity of a graph changes by relocating a connected branch from one vertex to another vertex, and then minimize the algebraic connectivity among all connected graphs of order $n$ with fixed domination number $γ\le \frac{n+2}{3}$, and finally present a lower bound for the algebraic connectivity in terms of the domination number.

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arXiv2013

The signature of line graphs and power trees

Let $G$ be a graph and let $A(G)$ be the adjacency matrix of $G$. The signature $s(G)$ of $G$ is the difference between the positive inertia index and the negative inertia index of $A(G)$. Ma et al. [Positive and negative inertia index of a graph, Linear Algebra and its Applications 438(2013)331-341] conjectured that $-c_3(G)\leq s(G)\leq c_5(G),$ where $c_3(G)$ and $c_5(G)$ respectively denote the number of cycles in $G$ which have length $4k+3$ and $4k+5$ for some integers $k \ge 0$, and proved the conjecture holds for trees, unicyclic or bicyclic graphs. It is known that $s(G)=0$ if $G$ is bipartite, and the signature is closely related to the odd cycles or nonbipartiteness of a graph from the existed results. In this paper we show that the conjecture holds for the line graph and power trees.

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arXiv2013

The Distance Coloring of Graphs

Let $G$ be a connected graph with maximum degree $Δ\ge 3$. We investigate the upper bound for the chromatic number $χ_γ(G)$ of the power graph $G^γ$. It was proved that $χ_γ(G) \leΔ\frac{(Δ-1)^γ-1}{Δ-2}+1=:M+1$ with equality if and only $G$ is a Moore graph. If $G$ is not a Moore graph, and $G$ holds one of the following conditions: (1) $G$ is non-regular, (2) the girth $g(G) \le 2γ-1$, (3) $g(G) \ge 2γ+2$, and the connectivity $κ(G) \ge 3$ if $γ\ge 3$, $κ(G) \ge 4$ but $g(G) >6$ if $γ=2$, (4) $Δ$ is sufficiently large than a given number only depending on $γ$, then $χ_γ(G) \le M-1$. By means of the spectral radius $λ_1(G)$ of the adjacency matrix of $G$, it was shown that $χ_2(G) \le λ_1(G)^2+1$, with equality holds if and only if $G$ is a star or a Moore graph with diameter 2 and girth 5, and $χ_γ(G) < λ_1(G)^γ+1$ if $γ\ge 3$.

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