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Let $K$ be a simplicial complex, and let $\Delta_i^{up}(K)$ be the $i$-th up normalized Laplacian of $K$. Horak and Jost showed that the largest eigenvalue of $\Delta_i^{up}(K)$ is at most $i+2$, and characterized the equality case by the orientable or non-orientable circuits. In this paper, by using the balancedness of signed graphs, we show that $\Delta_i^{up}(K)$ has an eigenvalue $i+2$ if and only if $K$ has an $(i+1)$-path connected component $K'$ such that the $i$-th signed incidence graph $B_i(K')$ is balanced, which implies Horak and Jost's characterization. We also characterize the multiplicity of $i+2$ as an eigenvalue of $\Delta_i^{up}(K)$, which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes $K$ with $\Delta_i^{up}(K)$ having an eigenvalue $i+2$ by using wedge, Cartesian product and duplication of motifs., Comment: arXiv admin note: substantial text overlap with arXiv:2405.19078

A graph $G$ is factored into graphs $H$ and $K$ via a matrix product if there exist adjacency matrices $A$, $B$, and $C$ of $G$, $H$, and $K$, respectively, such that $A = BC$. In this paper, we study the spectral aspects of the matrix product of graphs, including regularity, bipartiteness, and connectivity. We show that if a graph $G$ is factored into a connected graph $H$ and a graph $K$ with no isolated vertices, then certain properties hold. If $H$ is non-bipartite, then $G$ is connected. If $H$ is bipartite and $G$ is not connected, then $K$ is a regular bipartite graph, and consequently, $n$ is even. Furthermore, we show that trees are not factorizable, which answers a question posed by Maghsoudi et al., Comment: Comments are welcome

Let $K$ be a simplical complex, and let $\mathcal{L}_i^{up}(K), \mathcal{Q}_i^{up}(K)$ be the $i$-th up Laplacian and signless Laplacian of $K$, respectively. In this paper we proved that the largest eigenvalue of $\mathcal{L}_i^{up}(K)$ is not greater than the largest eigenvalue of $\mathcal{Q}_i^{up}(K)$; furthermore, if $K$ is $(i+1)$-path connected, then the equality holds if and only if the $i$-th incidence signed graph $B_i(K)$ of $K$ is balanced. As an application we provided an upper bound for the largest eigenvalue of the $i$-th up Laplacian of $K$, which improves the bound given by Horak and Jost and generalizes the result of Anderson and Morley on graphs.We characterized the balancedness of simplicial complexes under operations such as wedge sum, join, Cartesian product and duplication of motifs. For each $i \ge 0$, by using wedge sum or duplication of motifs, we can construct an infinitely many $(i+1)$-path connected simplicial complexes $K$ with $B_i(K)$ being balanced.

Let $F$ be a graph and let $\mathcal{B}_r(F)$ be the class of $r$-uniform Berge-$F$ hypergraphs. In this paper, by establishing a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure via walks, we give a spectral asymptotic bound for $\mathcal{B}_{r}(C_3)$-free linear $r$-uniform hypergraphs and upper bounds for the spectral radii of $\mathcal{B}_{r}(K_{2,t})$-free or $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs, where $C_{3}$ and $K_{s,t}$ are respectively the triangle and the complete bipartite graph with one part having $s$ vertices and the other part having $t$ vertices. Our work implies an upper bound for the number of edges of $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs, and extends some known work on (spectral) extreme problems of hypergraphs.

In this paper, by the representation theory of symmetric group, we give a decomposition of the Laplace operator (in matrix form) of a covering simplicial complex into the direct sum of some matrices, including the Laplace operator of the underlying simplicial complex. So, the spectrum of a covering simplicial complex can be expressed into a union of the spectrum of the underlying simplicial complex and the spectra of some other matrices, which implies a result of Horak and Jost. In particular, we show that the spectrum of a $2$-fold covering simplicial complex is the union the that of the underlying simplicial complex and that of an incidence-signed simplicial complex, which is an analog of Bilu and Linial's result on graphs. Finally we show that the dimension of the cohomology group of a covering simplicial complex is greater than or equal to that of the underlying simplicial complex.

In this paper we investigate Ramanujan hypergraphs by using hypergraph coverings. We first show that the spectrum of a $k$-fold covering $\bar{H}$ of a connected hypergraph $H$ contains the spectrum of $H$, and that it is the union of the spectrum of $H$ and the spectrum of an incidence-signed hypergraph with $H$ as underlying hypergraph if $k=2$, which generalizes Bilu-Linial result on graph coverings. We give a lower bound for the second largest eigenvalue of a $d$-regular hypergraph by universal cover, which generalizes Alon-Boppana bound on $d$-regular graphs and Feng-Li bound on $(d,r)$-regular hypergraphs. By using interlacing family of polynomials, we prove that every $(d,r)$-regular hypergraph has a right-sided Ramanujan $2$-covering, and has a left-sided Ramanujan $2$-covering if the roots of the matching polynomial of its incident graph satisfy some condition. By Ramanujan $2$-coverings, we prove the existence of some families of infinite many left-sided or right-sided $(d,r)$-regular Ramanujan hypergraphs under certain conditions on $d$ and $r$.

The edge blow-up of a graph $G$, denoted by $G^{p+1}$, is obtained by replacing each edge of $G$ with a clique of order $p+1$, where the new vertices of the cliques are all distinct. Yuan [J. Comb. Theory, Ser. B, 152 (2022) 379-398] determined the range of the Tur\'{a}n numbers for edge blow-up of all bipartite graphs and the exact Tur\'{a}n numbers for edge blow-up of all non-bipartite graphs. In this paper we prove that the graphs with the maximum spectral radius in an $n$-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when $n$ is sufficiently large., Comment: 22. arXiv admin note: text overlap with arXiv:2208.06550

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size $r-2$ disjoint from the vertex set of $F$ such that distinct edges are enlarged by disjoint subsets. Let $ex_{r}^{lin}(n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ be the maximum number of edges and the maximum spectral radius of all $F^{r}$-free linear $r$-uniform hypergraphs with $n$ vertices, respectively. In this paper, we present the sharp (or asymptotic) bounds of $ex_{r}^{lin}( n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ by establishing the connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where $F$ is a $(k+1)$-color critical graph or a graph with chromatic number $k$.

In this paper we investigate the traces of the adjacency tensor of hypergraphs (simply called the traces of hypergraphs). We give new expressions for the traces of hypertrees and linear unicyclic hypergraphs by the weight function assigned to their connected sub-hypergraphs, and provide some perturbation results for the traces of a hypergraph with cut vertices. As applications we determine the unique hypertree with maximum Estrada index among all hypertrees with fixed number of edges and perfect matchings, and the unique unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph with fixed number of edges and girth $3$.

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency matrix, and the spectrum of $G^m$ is referring to its adjacency tensor. The graph $G$ is called determined by high-ordered spectra (DHS for short) if, whenever $H$ is a graph such that $H^m$ is cospectral with $G^m$ for all $m$, then $H$ is isomorphic to $G$. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra.

Let $\h$ be a connected $k$-graph with maximum degree ${\Delta}\geq 2$ and let $\mu(\h, x)$ be the matching polynomial of $\h$. In this paper, we focus on studying the distribution of zeros of the matching polynomials of $k$-graphs. We prove that the zeros (with multiplicities) of $\mu(\h, x)$ are invariant under a rotation of an angle $2\pi/{\ell}$ in the complex plane for some positive integer $\ell$ and $k$ is the maximum integer with this property. Let $\lambda(\h)$ denote the maximum modulus of all zeros of $\mu(\h, x)$. We show that $\lambda(\h)$ is a simple root of $\mu(\h, x)$ and $$\Delta^{1\over k} \leq \lambda(\h)< \frac{k}{k-1}\big((k-1)(\Delta-1)\big)^{1\over k}.$$ To achieve these, we introduce the path tree $\T(\h,u)$ of $\h$ with respect to a vertex $u$ of $\h$, which is a $k$-tree, and prove that $$\frac{\mu(\h-u,x)}{\mu(\h, x)} = \frac{\mu(\T(\h,u)-u,x) }{\mu(\T(\h,u),x)},$$ which generalizes the celebrated Godsil's identity on the matching polynomial of graphs.

Let $\mathcal{H}$ be an $m$-uniform hypergraph, and let $\mathcal{A}(\mathcal{H})$ be the adjacency tensor of $\mathcal{H}$ which can be viewed as a system of homogeneous polynomials of degree $m-1$. Morozov and Shakirov generalized the traces of linear systems to nonlinear homogeneous polynomial systems and obtained explicit formulas for multidimensional resultants. Sun, Zhou and Bu introduced the Estrada index of uniform hypergraphs which is closely related to the traces of their adjacency tensors. In this paper we give formulas for the traces of $\mathcal{A}(\mathcal{H})$ when $\mathcal{H}$ contains cut vertices, and obtain results on the traces and Estrada index when $\mathcal{H}$ is perturbed under local changes. We prove that among all hypertrees with fixed number of edges, the hyperpath is the unique one with minimum Estrada index and the hyperstar is the unique one with maximum Estrada index.

Let $H$ be a connected $m$-uniform hypergraph, and let $\mathcal{A}(H)$ be the adjacency tensor of $H$ whose spectrum is simply called the spectrum of $H$. Let $s(H)$ denote the number of eigenvectors of $\mathcal{A}(H)$ associated with the spectral radius, and $c(H)$ denote the number of eigenvalues of $\mathcal{A}(H)$ with modulus equal to the spectral radius, which are respectively called the stabilizing index and cyclic index of $H$. Let $\bar{H}$ be a $k$-fold covering of $H$ which can be obtained from some permutation assignment in the symmetric group $\mathbf{S}_k$ on $H$. In this paper, we first characterize the connectedness of $\bar{H}$ by its incidence graph and the permutation assignment, and then investigate the relationship between the spectral property of $H$ and that of $\bar{H}$. By applying module theory and group representation, if $\bar{H}$ is connected, we prove that $s(H) \mid s(\bar{H})$ and $c(H) \mid c(\bar{H})$. In particular, when $\bar{H}$ is a $2$-fold covering of $H$, if $m$ is even, we show that regardless of multiplicities, the spectrum of $\bar{H}$ contains the spectrum of $H$ and the spectrum of a signed hypergraph with $H$ as underlying hypergraph; if $m$ is odd, we give an explicit formula for $s(\bar{H})$. We also find some differences on the spectral property between hypergraph coverings and graph coverings by examples.

An oriented hypergraph is a hypergraph together with an incidence orientation such that each edge-vertex incidence is given a label of $+1$ or $-1$. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors, and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs.

Let $G$ be a connected graph and let $T$ be a spanning tree of $G$. A partial orientation $\sigma$ of $G$ respect to $T$ is an orientation of the edges of $G$ except those edges of $T$, the resulting graph associated with which is denoted by $G_T^\sigma$. In this paper we prove that there exists a partial orientation $\sigma$ of $G$ respect to $T$ such that the largest eigenvalue of the Hermitian adjacency matrix of $G_T^\sigma$ is at most the largest absolute value of the roots of the matching polynomial of $G$.

Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are more close to bipartite simple graphs from the viewpoint of both structure and spectrum. A hypergraph is called minimal non-odd-transversal if it is non-odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

Let $G$ be connected uniform hypergraph and let $\mathcal{A}(G)$ be the adjacency tensor of $G$. The stabilizing index of $G$ is the number of eigenvectors of $\mathcal{A}(G)$ associated with the spectral radius, and the cyclic index of $G$ is the number of eigenvalues of $\mathcal{A}(G)$ with modulus equal to the spectral radius. Let $G_1 \odot G_2$ and $G_1 \Box G_2$ be the coalescence and Cartesian product of connected $m$-uniform hypergraphs $G_1$ and $G_2$ respectively. In this paper, we give explicit formulas for the the stabilizing indices and cyclic indices of $G_1 \odot G_2$ and $G_1 \Box G_2$ in terms of those of $G_1$ and $G_2$ or the invariant divisors of their incidence matrices over $\mathbb{Z}_m$, respectively.

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 $\mathcal{A}(G)$ be the adjacency tensor of $G$. The least H-eigenvalue of $\mathcal{A}(G)$ refers to the least real eigenvalue of $\mathcal{A}(G)$ associated with a real eigenvector. In this paper we obtain a perturbation result on the least H-eigenvalue of $\mathcal{A}(G)$ 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., Comment: arXiv admin note: substantial text overlap with arXiv:1902.04233

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

Let $\Gamma^{(x_0)}$ be a finite rooted tree, for which $\Gamma$ is the underlying tree and $x_0$ the root. Let $T$ be the Terwilliger algebra of $\Gamma$ with respect to $x_0$. We study the structure of the principal $T$-module. As a result, it is shown that $T$ recognizes the isomorphism class of $\Gamma^{(x_0)}$., Comment: 11 pages

Let $G$ be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of $G$ is simply called the least eigenvalue of $G$ and the corresponding H-eigenvectors are called the first eigenvectors of $G$. In this paper we give some numerical and structural properties about the first eigenvectors of $G$ which contains an odd-bipartite branch, and investigate how the least eigenvalue of $G$ changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite 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.

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 $\mathbb{Z}_m$., Comment: arXiv admin note: text overlap with arXiv:1704.08799

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 $G$.

By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike hypergraphs including hyperpaths and hyperstars,which are given recursively and explicitly.

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., Comment: arXiv admin note: text overlap with arXiv:1707.07414

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.

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.

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.

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:1408.3303

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.

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.

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.

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.

In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic connectivity in terms of the matching number or edge covering number., Comment: arXiv admin note: substantial text overlap with arXiv:1310.8533

An automorphism of a graph $G=(V,E)$ is a bijective map $\phi$ from $V$ to itself such that $\phi(v_i)\phi(v_j)\in E$ $\Leftrightarrow$ $v_i v_j\in E$ 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 $\sigma$ 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 $\sigma\boldsymbol{u}=(u_{\sigma^{-1}1},u_{\sigma^{-1}2},\ldots,u_{\sigma^{-1} n})^t$, so $\sigma$ 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.,} $\sigma$ is an automorphism of $G$ if and only if every eigenspace of $\mathbf{A}(G)$ is $\sigma$-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 $\lambda$, $\mathrm{span}(\mathfrak{G}\boldsymbol{v})$ is a subspace of the eigenspace $V_{\lambda}$. 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 $\mathbf{A}(G)$., Comment: 20 pages, 3 figures

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 $\gamma \le \frac{n+2}{3}$, and finally present a lower bound for the algebraic connectivity in terms of the domination number.

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.

Let $G$ be a connected non-bipartite graph on $n$ vertices with domination number $\gamma \le \frac{n+1}{3}$. We investigate the least eigenvalue of the signless Laplacian of $G$, and present a lower bound for such eigenvalue in terms of the domination number $\gamma$., Comment: arXiv admin note: text overlap with arXiv:1310.8533

How to construct a suitable measurement matrix is still an open question in compressed sensing. A significant part of the recent work is that the measurement matrices are not completely random on the entries but exhibit considerable structure. In this paper, we proved that the symmetric Toeplitz matrix and its transforms can be used as measurement matrix and recovery signal with high probability. Compared with random matrices (e.g. Gaussian and Bernullio matrices) and some structured matrices (e.g. Toeplitz and circulant matrices), we need to generate fewer independent entries to obtain the measurement matrix while the effectiveness of recovery does not get worse. Furthermore, the signal can be recovered more efficiently by the algorithm.

Finding a suitable measurement matrix is an important topic in compressed sensing. Though the known random matrix, whose entries are drawn independently from a certain probability distribution, can be used as a measurement matrix and recover signal well, in most cases, we hope the measurement matrix imposed with some special structure. In this paper, based on random graph models, we show that the mixed symmetric random matrices, whose diagonal entries obey a distribution and non-diagonal entries obey another distribution, can be used to recover signal successfully with high probability., Comment: 10 pages, 3 figures. arXiv admin note: text overlap with arXiv:1212.3799

A graph in a certain graph class is called minimizing if the least eigenvalue of the adjacency matrix of the graph attains the minimum among all graphs in that class. Bell {\it et al.} have characterized the minimizing graphs in the class of connected graphs of order $n$ and size $m$, whose complements are either disconnected or contain a clique of order at least $n/2$. In this paper we discuss the minimizing graphs of a special class of graphs of order $n$ whose complements are connected and contains exactly one cycle (namely the the class $\mathscr {U}^{c}_{n}$ of graphs whose complements are unicyclic), and characterize the unique minimizing graph in $\mathscr {U}^{c}_{n}$ when $n \geq 20$.

The rank of a graph G is defined to be the rank of its adjacency matrix A(G). In this paper we characterize all connected triangle-free graphs with rank 6.

The task of compressed sensing is to recover a sparse vector from a small number of linear and non-adaptive measurements, and the problem of finding a suitable measurement matrix is very important in this field. While most recent works focused on random matrices with entries drawn independently from certain probability distributions, in this paper we show that a partial random symmetric Bernoulli matrix whose entries are not independent, can be used to recover signal from observations successfully with high probability. The experimental results also show that the proposed matrix is a suitable measurement matrix., Comment: arXiv admin note: text overlap with arXiv:0902.4394 by other authors

Let $G$ be a connected graph with maximum degree $\Delta \ge 3$. We investigate the upper bound for the chromatic number $\chi_\gamma(G)$ of the power graph $G^\gamma$. It was proved that $\chi_\gamma(G) \le\Delta\frac{(\Delta-1)^{\gamma}-1}{\Delta-2}+1=:M+1$ 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 $g(G) \le 2\gamma-1$, (3) $g(G) \ge 2\gamma+2$, and the connectivity $\kappa(G) \ge 3$ if $\gamma \ge 3$, $\kappa(G) \ge 4$ but $g(G) >6$ if $\gamma =2$, (4) $\Delta$ is sufficiently large than a given number only depending on $\gamma$, then $\chi_\gamma(G) \le M-1$. By means of the spectral radius $\lambda_1(G)$ of the adjacency matrix of $G$, it was shown that $\chi_2(G) \le \lambda_1(G)^2+1$, with equality holds if and only if $G$ is a star or a Moore graph with diameter 2 and girth 5, and $\chi_\gamma(G) < \lambda_1(G)^\gamma+1$ if $\gamma \ge 3$.

The Wiener polarity index of a graph G is the number of unordered pairs of vertices u, v such that the distance between u and v is 3. In this paper we give an explicit formula for the Wiener polarity index of cactus graphs. We also deduce formulas for some special cactus graphs.

In this paper we determine the graph whose least eigenvalue of signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and an upper bound for the least eigenvalue of signless Laplacian of a graph in terms of the number of pendent vertices.