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In this paper, we study the spectral radius of the clique tensor A(G) associated with a graph G. This tensor is a higher-order extensions of the adjacency matrix of G. A lower bound of the clique number is given via the spectral radius of A(G). It is an extension of Nikiforov's spectral bound and tighter than the bound of Nikiforov in some classes of graphs. Furthermore, we obtain a spectral version of the Erdos-Simonovits stability theorem for clique tensors based on this bound.

Let $G$ be a connected graph and let $F$ be a connected subgraph of $G$ with a given structure. We consider that the centrality of a vertex $i$ of $G$ is determined by the centrality of other vertices in all subgraphs contain $i$ and isomorphic to $F$. In this paper we propose an $F$-subgraph tensor and an $F$-subgraph eigenvector centrality of $G$. When the graph is $F$-connected, we show that the $F$-subgraph tensor is weakly irreducible, and in this case, the $F$-subgraph eigenvector centrality exists. Specifically, when we choose $F$ to be a path $P_1$ of length $1$(or a complete graph $K_2$), the $F$-eigenvector centrality is eigenvector centrality of $G$. Furthermore, we propose the $(K_2,F)$-subgraph eigenvector centrality of $G$ and prove it always exists when $G$ is connected. Specifically, the $P_2$-subgraph eigenvector centrality and $(K_2,F)$-subgraph eigenvector centrality are studied. Some examples show that the ranking of vertices under them differs from the rankings under several classic centralities. Vertices of a regular graph have the same eigenvector centrality scores. But the $(K_2,K_3)$-subgraph eigenvector centrality can distinguish vertices in a given regular graph., Comment: 20 pages, 5 figures

Erd\H{o}s-Stone Theorem is a well-known result in extremal graph theory which determines the asymptotic behaviour of maximum number of edges in an $n$-vertex $H$-free graph. In 2009, Nikiforov gave a spectral version of Erd\H{o}s-Stone Theorem. In this paper, we obtain a tensor's spectral version of Erd\H{o}s-Stone Theorem.

The $d$-th order Laplacian spectral moment of a $k$-uniform hypergraph is the sum of the $d$-th powers of all eigenvalues of its Laplacian tensor. In this paper, we obtain some expressions of the Laplacian spectral moments for $k$-uniform power hypergraphs, and these expressions can be represented by some parameters of graphs. And we show that some graphs can be determined by their high-order Laplacian spectrum by using the Laplacian spectral moments of power hypergraphs.

In this paper, the Laplacian characteristic polynomial of uniform hypergraphs with cut vertices or pendant edges and the Laplacian matching polynomial of uniform hypergraphs are characterized.The multiplicity of the zero Laplacian eigenvalue of uniform hypertrees is given, which proves the conjecture in \cite{zheng2023zero} (The zero eigenvalue of the Laplacian tensor of a uniform hypergraph, Linear and Multilinear Algebra, (2023) Doi:10.1080/03081087.2023.2172541).

The lexicographical ordering of hypergraphs via spectral moments is called the $S$-order of hypergraphs.In this paper, the $S$-order of hypergraphs is investigated.We characterize the first and last hypergraphs in an $S$-order of all uniform hypertrees and all linear unicyclic uniformhypergraphs with given girth, respectively. And we give the last hypergraph in an $S$-order of all linear unicyclic uniform hypergraphs.

By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions. The biclique partition formula also extends the results of Knuth and Levine from line digraphs to general digraphs.

It is well-known that the spectral radius of a connected uniform hypergraph is an eigenvalue of the hypergraph. However, its algebraic multiplicity remains unknown. In this paper, we use the Poisson Formula and matching polynomials to determine the algebraic multiplicity of the spectral radius of a uniform hypertree.

The $k$-power hypergraph $G^{(k)}$ is the $k$-uniform hypergraph that is obtained by adding $k-2$ new vertices to each edge of a graph $G$, for $k \geq 3$. A parity-closed walk in $G$ is a closed walk that uses each edge an even number of times. In an earlier paper, we determined the eigenvalues of the adjacency tensor of $G^{(k)}$ using the eigenvalues of signed subgraphs of $G$. Here, we express the entire spectrum (that is, we determine all multiplicities and the characteristic polynomial) of $G^{(k)}$ in terms of parity-closed walks of $G$. Moreover, we give an explicit expression for the multiplicity of the spectral radius of $G^{(k)}$. Our results are mainly obtained by exploiting the so-called trace formula to determine the spectral moments of $G^{(k)}$. As a side result, we show that the number of parity-closed walks of given length is the corresponding spectral moment averaged over all signed graphs with underlying graph $G$. We also extrapolate the characteristic polynomial of $G^{(k)}$ to $k=2$, thereby introducing a pseudo-characteristic function. Among other results, we show that this function is the geometric mean of the characteristic polynomials of all signed graphs on $G$ and characterize when it is a polynomial. This supplements a result by Godsil and Gutman that the arithmetic mean of the characteristic polynomials of all signed graphs on $G$ equals the matching polynomial of $G$.

Mantel's theorem is a classical result in extremal graph theory which implies that the maximum number of edges of a triangle-free graph of order $n$. In 1970, E. Nosal obtained a spectral version of Mantel's theorem which gave the maximum spectral radius of a triangle-free graph of order $n$. In this paper, the clique tensor of a graph $G$ is proposed and the spectral Mantel's theorem is extended via the clique tensor. Furthermore, a sharp upper bound of the number of cliques in $G$ via the spectral radius of the clique tensor is obtained. And we show that the results of this paper implies that a result of Erd\H{o}s [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 7 (1962)] under certain conditions.

We show how all eigenvalues of a power hypergraph $G^{(k)}$ can be generated from the eigenvalues of signed subgraphs of the underlying graph $G$. This fixes an incorrect statement in the case of power hypergraphs from [Linear Algebra and its Applications, 590:243-257, 2020].

The spectrum of the $k$-power hypergraph of a graph $G$ is called the $k$-ordered spectrum of $G$.If graphs $G_1$ and $G_2$ have same $k$-ordered spectrum for all positive integer $k\geq2$, $G_1$ and $G_2$ are said to be high-ordered cospectral. If all graphs who are high-ordered cospectral with the graph $G$ are isomorphic to $G$, we say that $G$ is determined by the high-ordered spectrum.In this paper, we use the high-ordered spectrum of graphs to study graph isomorphism and show that all Smith's graphs are determined by the high-ordered spectrum.And we give infinitely many pairs of trees with same spectrum but different high-ordered spectrum by high-ordered cospectral invariants of trees,it means that we can determine that these cospectral trees are not isomorphism by the high-ordered spectrum.

Uniform hypergraphs have a natural one-to-one correspondence to tensors. In this paper, we investigate the Estrada index and subgraph centrality of an $m$-uniform hypergraph $\mathcal{H}$ via the adjacency tensor. We establish some bounds for the Estrada index and give expressions of the subgraph centrality in terms of graph parameters of the multi-digraphs associated with $\mathcal{H}$. When $\mathcal{H}$ is $2$-uniform, the above Estrada index and subgraph centrality are the Estrada index and subgraph centrality of a graph.

Let $\mathcal{G}$ be a $k$-uniform hypergraph, $\mathcal{L}_{\mathcal{G}}$ be its Laplacian tensor. And $\beta( \mathcal{G})$ denotes the maximum number of linearly independent nonnegative eigenvectors of $\mathcal{L}_{\mathcal{G}}$ corresponding to the eigenvalue $0$. In this paper, $\beta( \mathcal{G})$ is called the geometry connectivity of $\mathcal{G}$. We show that the number of connected components of $\mathcal{G}$ equals the geometry connectivity $\beta( \mathcal{G})$.

For a simple, undirected and connected graph $G$, $D_{\alpha}(G) = \alpha Tr(G) + (1-\alpha) D(G)$ is called the $\alpha$-distance matrix of $G$, where $\alpha\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $\alpha$-distance energy of $G$ was defined based on the spectra of $D_{\alpha}(G)$. In this paper, we define the $\alpha$-distance Estrada index of $G$ in terms of the eigenvalues of $D_{\alpha}(G)$. And we give some bounds on the spectral radius of $D_{\alpha}(G)$, $\alpha$-distance energy and $\alpha$-distance Estrada index of $G$.

In this paper, we give a reduced formula of the characteristic polynomial of $k$-uniform hypergraphs with a pendant edge. And the explicit characteristic polynomial and all distinct eigenvalues of $k$-uniform hyperpath are given.

In this paper, we show that a uniform hypergraph $\mathcal{G}$ is connected if and only if one of its inverse Perron values is larger than $0$. We give some bounds on the bipartition width, isoperimetric number and eccentricities of $\mathcal{G}$ in terms of inverse Perron values. By using the inverse Perron values, we give an estimation of the edge connectivity of a $2$-design, and determine the explicit edge connectivity of a symmetric design. Moreover, relations between the inverse Perron values and resistance distance of a connected graph are presented.

In this paper, we give some bounds for principal eigenvector and spectral radius of connected uniform hypergraphs in terms of vertex degrees, the diameter, and the number of vertices and edges.

In this paper, we give the Minc-type bound for spectral radius of nonnegative tensors. We also present the bounds for the spectral radius and the eigenvalue inclusion sets of the general product of tensors.

In this paper, we investigate spectral properties of the adjacency tensor, Laplacian tensor and signless Laplacian tensor of general hypergraphs (including uniform and non-uniform hypergraphs). We obtain some bounds for the spectral radius of general hypergraphs in terms of vertex degrees, and give spectral characterizations of odd-bipartite hypergraphs.

Let $G$ be a connected uniform hypergraphs with maximum degree $\Delta$, spectral radius $\lambda$ and minimum H-eigenvalue $\mu$. In this paper, we give some lower bounds for $\Delta-\lambda$, which extend the result of [S.M. Cioab\u{a}, D.A. Gregory, V. Nikiforov, Extreme eigenvalues of nonregular graphs, J. Combin. Theory, Ser. B 97 (2007) 483-486] to hypergraphs. Applying these bounds, we also obtain a lower bound for $\Delta+\mu$.

For a nonnegative weakly irreducible tensor $\mathcal{A}$, we give some characterizations of the spectral radius of $\mathcal{A}$, by using the digraph of tensors. As applications, some bounds on the spectral radius of the adjacency tensor and the signless Laplacian tensor of the $k$-uniform hypergraphs are shown.

Let $\mathcal{A}$ be an order $t$ dimension $m\times n\times \cdots \times n$ tensor over complex field. In this paper, we study some {generalized inverses} of $\mathcal{A}$, the {$k$-T-idempotent tensors} and the idempotent tensors based on the general tensor product. Using the tensor generalized inverse, some solutions of the equation $\mathcal{A}\cdot x^{t-1}=b$ are given, where $x$ and $b$ are dimension $n$ and $m$ vectors, respectively. The {generalized inverses} of some block tensors, the eigenvalues of {$k$-T-idempotent tensors} and idempotent tensors are given. And the relation between the generalized inverses of tensors and the $k$-T-idempotent tensors is also showed.

In this paper, we define the minimum (maximum) rank, term rank and the sign nonsingular of tensors. The sufficiency and necessity for the minimum rank of a real tensor to be $1$ is given. And we show that the maximum rank of a tensor is not less than the term rank. We also prove that the minimum rank of a sign nonsingular tensor is not less than the dimension of it. And we get some characterizations of a tensor having sign left or sign right inverses.

For a $k$-uniform hypergraph $H$, we obtain some trace formulas for the Laplacian tensor of $H$, which imply that $\sum_{i=1}^nd_i^s$ ($s=1,\ldots,k$) is determined by the Laplacian spectrum of $H$, where $d_1,\ldots,d_n$ is the degree sequence of $H$. Using trace formulas for the Laplacian tensor, we obtain expressions for some coefficients of the Laplacian polynomial of a regular hypergraph. We give some spectral characterizations of odd-bipartite hypergraphs, and give a partial answer to a question posed by Shao et al \cite{ShaoShanWu}. We also give some spectral properties of power hypergraphs, and show that a conjecture posed by Hu et al \cite{HuQiShao} holds under certain conditons.

Two $k$-uniform hypergraphs are said to be cospectral (E-cospectral), if their adjacency tensors have the same characteristic polynomial (E-characteristic polynomial). A $k$-uniform hypergraph $H$ is said to be determined by its spectrum, if there is no other non-isomorphic $k$-uniform hypergraph cospectral with $H$. In this note, we give a method for constructing E-cospectral hypergraphs, which is similar with Godsil-McKay switching. Some hypergraphs are shown to be determined by their spectra.