Spectral classes of hypergraphs

在线条之外着色：广义超图着色的谱界

Coloring outside the lines: Spectral bounds for generalized hypergraph colorings

本文提出了一种广义超图着色的谱界，探讨了d-正色和d-反正色着色的谱界限及其条件。

Spectral properties of general hypergraphs

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.

Axiomatisability and hardness for universal Horn classes of hypergraphs

We characterise finite axiomatisability and intractability of deciding membership for universal Horn classes generated by finite loop-free hypergraphs.

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.

The first few unicyclic and bicyclic hypergraphs with larger spectral radii

A connected $k$-uniform hypergraph with $n$ vertices and $m$ edges is called $r$-cyclic if $n=m(k-1)-r+1$. For $r=1$ or $2$, the hypergraph is simply called unicyclic or bicyclic. In this paper we investigate hypergraphs that attain larger spectral radii among all simple connected $k$-uniform unicyclic and bicyclic hypergraphs. Specifically, by using some edge operations, the formula on power hypergraph eigenvalues, the weighted incidence matrix and a result on linear unicyclic hypergraphs, we determined the first five hypergraphs with larger spectral radius among all unicyclic hypergraphs and…

超图的谱理论：综述

Spectral Theory of Hypergraphs: A Survey

本文综述了基于超矩阵框架的超图谱理论，系统梳理了超图谱性质、谱半径界、Turán型问题谱版本及特征多项式与匹配多项式关系等核心内容，为简化超图谱计算提供了理论支撑。

Computing the $p$-Spectral Radii of Uniform Hypergraphs with Applications

The $p$-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the $p$-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius…

超图乘积的特征值与互为特征值性质

Eigenvalues of Hypergraph Products and Reciprocal Eigenvalue Property

本文研究了超图的连接、克罗内克和冠状乘积的邻接、拉普拉斯和无向拉普拉斯特征值，探讨了这些谱在操作下的行为，并扩展了互为特征值性质到超图，证明幂超树不满足该性质。

双曲超图谱矩的字典序排序

Lexicographical ordering by spectral moments of bicyclic hypergraphs

本文研究双曲超图的谱矩字典序排序，确定了给定围长与边数的线性双曲一致超图在S-序下的首尾超图，为超图谱序研究提供新结果。

Connected Hypergraphs with Small Spectral Radius

In 1970 Smith classified all connected graphs with the spectral radius at most $2$. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to $r$-uniform hypergraphs. In this paper, we generalize the Smith's theorem to $r$-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected $r$-uniform hypergraphs is $ρ_r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected $r$-uniform hypergraphs with …

On the $α$-spectral radius of uniform hypergraphs

For $0\leα<1$ and a uniform hypergraph $G$, the $α$-spectral radius of $G$ is the largest $H$-eigenvalue of $α\mathcal{D}(G) +(1-α)\mathcal{A}(G)$, where $\mathcal{D}(G)$ and $\mathcal{A}(G)$ are the diagonal tensor of degrees and the adjacency tensor of $G$, respectively. We give upper bounds for the $α$-spectral radius of a uniform hypergraph, propose some transformations that increase the $α$-spectral radius, and determine the unique hypergraphs with maximum $α$-spectral radius in some classes of uniform hypergraphs.

On the spectral radii of the unicyclic hypergraphs with fixed matching number

We determine the unique hypergraphs with maximum spectral radius among all connected $k$-uniform ($k\geq 3$) unicyclic hypergraphs with matching number at least $z$, and among all connected $k$-uniform ($k\geq 3$) unicyclic hypergraphs with a given matching number, respectively.

Cops and Robber on Cartesian products and some classes of hypergraphs

The game of Cops and Robber is a pursuit-evasion game which is usually played on a connected graph. In the game, a set of cops and a robber move around the vertices of a graph along edges, where the cops aim to capture the robber, while the robber aims to avoid capture. Much research about this game have been done since the early 1980s. The game has a natural generalisation to being played on connected hypergraphs, where the cops and the robber may now move along hyperedges. In this paper, we shall provide a characterisation of all hypergraphs where one cop is sufficient to capture the robber.…

通过谱矩对超图进行字典序排序

Lexicographical ordering of hypergraphs via spectral moment

本文研究超图基于谱矩的字典序（S-序），刻画了给定围长的一致超树及线性单圈一致超图在该序中的首尾超图，并确定所有线性单圈一致超图中的末位超图。

The bounds of the spectral radius of general hypergraphs in terms of clique number

The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.

Spectral radius of uniform hypergraphs and degree sequences

In this paper, we present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.

A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs

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 …

Some results on the spectral radii of uniform hypergraphs

Let A(G) be the adjacency tensor (hypermatrix) of uniform hypergraph G. The maximum modulus of the eigenvalues of A(G) is called the spectral radius of G. In this paper, the conjecture of Fan et al. in [5] related to compare the spectral radii of some three uniform hypergraphs is solved. Moreover, some eigenvalues properties of a kind of uniform hypergraphs are obtained.

Distance spectral radius of uniform hypergraphs

We study the effect of three types of graft transformations to increase or decrease the distance spectral radius of uniform hypergraphs, and we determined the unique $k$-uniform hypertrees with maximum, second maximum, minimum and second minimum distance spectral radius, respectively.

Principal eigenvector and spectral radius of uniform hypergraphs

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.

Spectral Extremal Results for Hypergraphs

Let $F$ be a graph. A hypergraph is called Berge $F$ if it can be obtained by replacing each edge in $F$ by a hyperedge containing it. Given a family of graphs $\mathcal{F}$, we say that a hypergraph $H$ is Berge $\mathcal{F}$-free if for every $F \in \mathcal{F}$, the hypergraph $H$ does not contain a Berge $F$ as a subhypergraph. In this paper we investigate the connections between spectral radius of the adjacency tensor and structural properties of a linear hypergraph. In particular, we obtain a spectral version of Turán-type problems over linear $k$-uniform hypergraphs by using spectral me…

On the spectral radius of some linear hypergraphs

Here we study the spectral radii of some linear hypergraphs, that is, the maximum moduli of the eigenvalues of their corresponding adjacency matrices. We determine the hypertrees having the largest to seventh-largest spectral radii. The hypertrees with the largest and the second-largest spectral radii among all those with a given diameter are identified here. Unicyclic hypergraphs with a fixed cycle length having the largest, the second-largest, and the third-largest spectral radii are also determined. Furthermore, we also find which bicyclic and tricyclic linear hypergraphs have the largest a…

On some properties of the $α$-spectral radius of the $k$-uniform hypergraph

In this paper we show how the $α$-spectral radius changes under the edge grafting operations on connected $k$-uniform hypergraphs. We characterize the extremal hypertree for $α$-spectral radius among $k$-uniform non-caterpillar hypergraphs with given order, size and diameter. we also characerize the second largest $α$-spectral radius among all $k$-uniform supertrees on $n$ vertices by two methods.

有向超图的低差异与谱间隙：基于张量谱正则引理

Low discrepancy and spectral gap for directed hypergraphs via spectral regularity lemma for tensors

本文证明有向均匀超图的低差异等价于谱间隙，通过建立基于变分特征值的Frieze-Kannan型张量谱正则引理，给出Lenz-Mubayi定理的简短证明，并讨论相关变体。

非退化超图的谱Turán问题

Spectral Turán problems for nondegenerate hypergraphs

本研究拓展了Keevash–Lenz–Mubayi的超图谱极值判定准则，建立非退化超图的谱稳定性结果，推导出两类超图的通用谱Turán定理，并确定了多类超图的最大α-谱半径及对应极值结构。

幂超图的拉普拉斯谱矩

The Laplacian spectral moments of power hypergraphs

该研究推导了k-一致幂超图拉普拉斯谱矩的图参数表达式，并借此证明部分图可由高阶拉普拉斯谱确定，为超图谱理论与图的谱确定性研究提供了新方法。

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs

A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of $1$-Spernerness, thresholdness, and $2$-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of $1$-Sperner hypergraphs to …

Spectral Turán Type Problems on Cancellative Hypergraphs

Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric difference of any two edges is not contained in a third one. Equivalently, a $3$-uniform hypergraph $G$ is cancellative if and only if $G$ is $\{F_4, F_5\}$-free, where $F_4 = \{abc, abd, bcd\}$ and $F_5 = \{abc, abd, cde\}$. A classical result in extremal combinatorics stated that the maximum size of a cancellative hypergraph is achieved by the balanced complete tripartite $3$-uniform hypergraph, which was firstly proved by Bollobás and later by Keevash and Mubayi. In this paper, we consider spectral extremal problems for…

Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs

We give sharp upper bounds for the ordinary spectral radius and signless Laplacian spectral radius of a uniform hypergraph in terms of the average $2$-degrees or degrees of vertices, respectively, and we also give a lower bound for the ordinary spectral radius. We also compare these bounds with known ones.

Signless Normalized Laplacian for Hypergraphs

The spectral theory of the normalized Laplacian for chemical hypergraphs is further investigated. The signless normalized Laplacian is introduced and it is shown that its spectrum for classical hypergraphs coincides with the spectrum of the normalized Laplacian for bipartite chemical hypergraphs. Furthermore, the spectra of special families of hypergraphs are established.

Spectral bounds for non-uniform hypergraphs using weighted clique expansion

Hypergraphs are an invaluable tool to understand many hidden patterns in large data sets. Among many ways to represent hypergraph, one useful representation is that of weighted clique expansion. In this paper, we consider this representation for non-uniform hypergraphs. We generalize the spectral results for uniform hypergraphs to non-uniform hypergraphs and show that they extend in a natural way. We provide a bound on the largest eigenvalue with respect to the average degree of neighbours of a vertex in a graph. We also prove an inequality on the boundary of a vertex set in terms of the large…

图与超图局部结构的谱特征

Spectral characterizations of local structures of graphs and hypergraphs

本文研究图与超图局部结构的谱特征，揭示谱半径与局部子图的关系，并提出基于张量特征值的局部向量着色数谱界。

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 connecte…

图和偶均匀超图独立数的谱界

Spectral bounds for the independence number of graphs and even uniform hypergraphs

本文研究了图和偶均匀超图的独立数的谱界，扩展了霍夫曼界到偶均匀超图，并提出了确定图的独立数、香农容量和洛vasz数的简单谱条件。

On the spectral radius of nonregular uniform hypergraphs

Let $G$ be a connected uniform hypergraphs with maximum degree $Δ$, spectral radius $λ$ and minimum H-eigenvalue $μ$. In this paper, we give some lower bounds for $Δ-λ$, which extend the result of [S.M. Cioabă, 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 $Δ+μ$.

The smallest spectral radius of bicyclic uniform hypergraphs with a given size

Identifying graphs with extremal properties is an extensively studied topic in spectral graph theory. In this paper, we study the log-concavity of a type of iteration sequence related to the $α$-normal weighted incidence matrices which is presented by Lu and Man for computing the spectral radius of hypergraphs. By using results obtained about the sequence and the method of some edge operations, we will characterize completely extremal k-graphs with the smallest spectral radius among bicyclic hypergraphs with given size.

The extremal $p$-spectral radius of Berge-hypergraphs

Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $ϕ: E(G)\to E(H)$ such that $e\subseteq ϕ(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the $p$-spectral radius $λ^{(p)}(H)$ of $H$ is defined as \[ λ^{(p)}(H):=\max_{{\bf x}\in\mathbb{R}^n,\,\|{\bf x}\|_p=1} r\sum_{\{i_1,i_2,\ldots,i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r}. \] In this paper, we study the $p$-spectral radius of Berge-$G$ hypergraphs. We determine the $3$-uniform hypergraphs with maximum $p$-spectral radius for $p\geq 1$ among Berge-$G$ hypergraph…

谱超图理论的新矩阵，I

New matrices for spectral hypergraph theory, I

本文提出超图的统一矩阵，证明其与关联图邻接矩阵相同，从而利用其谱连接超图与图的结构性质，并引入相关不变量与特征值关联。

The p-spectral radius of k-partite and k-chromatic uniform hypergraphs

We characterize the r-graph with maximal p-spectral radius among the k-partite r-graphs of order n, and the 3-graph with maximal p-spectral radius among the k-chromatic 3-graphs of order n.

Spectral Properties of Oriented Hypergraphs

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the s…

Some spectral properties of uniform hypergraphs

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 hypergr…

Spectral theory of weighted hypergraphs via tensors

One way to study an hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eingenvalues with some techniques from numerical algebraic geometry.

On ABC spectral radius of uniform hypergraphs

Given a $k$-uniform hypergraph $G$ with vertex set $[n]$ and edge set $E(G)$, the ABC tensor $\mathcal{ABC}(G)$ of $G$ is the $k$-order $n$-dimensional tensor with \[ \mathcal{ABC}(G)_{i_1, \dots, i_k}= \begin{cases} \dfrac{1}{(k-1)!}\sqrt[k]{\dfrac{\sum_{i\in e}d_{i}-k}{\prod_{i\in e}d_{i}}} & \mbox{if $e\in E(G)$} 0 & \mbox{otherwise} \end{cases} \] for $i_j\in [n]$ with $j\in [k]$, where $d_i$ is the degree of vertex $i$ in $G$. The ABC spectral radius of a uniform hypergraph is the spectral radius of its ABC tensor. We give tight lower and upper bounds for the ABC spectra radius, and deter…

Lower bounds for the $\mathcal{A}_α$-spectral radius of uniform hypergraphs

For $0\leq α< 1$, the $\mathcal{A}_α$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_α(G):=α\mathcal{D}(G)+(1-α) \mathcal{A}(G)$, where $\mathcal{D}(G)$ and $A(G)$ are diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the $\mathcal{A}_α$-spectral radius and an average degree $\frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the …

The maximal spectral radius of the uniform unicyclic hypergraph with perfect matchings

Let $\mathcal{U}(n,k)$ and $Γ(n,k)$ be the set of the $k$-uniform linear and nonlinear unicyclic hypergraphs having perfect matchings with $n$ vertices respectively, where $n\geq k(k-1)$ and $k\geq 3$. By using some techniques of transformations and constructing the incidence matrices for the hypergraphs considered, we get the hypergraphs with the maximal spectral radii among three kinds of hypergraphs, namely $\mathcal{U}(n,k)$ with $n= 2k(k-1)$ and $n\geq 9k(k-1)$, $Γ(n,k)$ with $n\geq k(k-1)$, and $\mathcal{U}(n,k)\cup Γ(n,k)$ with $n\geq 2k(k-1)$, where $k\geq 3$.

Spectral extremal problems for hypergraphs

In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from `strong stability' forms of the corresponding (pure) extremal results. These results hold for the α-spectral radius defined using the α-norm for any α>1; the usual spectrum is the case α=2. Our results imply that any hypergraph Turán problem which has the stability property and whose extremal construction satisfies some rather mild continuity assumptions admits a corresponding spectral result. A particular example is to determine the maximum α-spectral…

Maximum spectral radius of outerplanar 3-uniform hypergraphs

In this paper, we study the maximum spectral radius of outerplanar $3$-uniform hypergraphs. Given a hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is a graph $G$ with $V(G)= V(\mathcal{H})$ and $E(G) = \{uv: uv \in h \textrm{ for some } h\in E(\mathcal{H})\}$. A graph is \textit{outerplanar} if it can be embedded in the plane such that all its vertices lie on the outer face. A $3$-uniform hypergraph $\mathcal{H}$ is called \textit{outerplanar} if its shadow has an outerplanar embedding such that every hyperedge of $\mathcal{H}$ is the vertex set of an interior triangular face of the sha…

关于$k$-一致超图的Estrada指数与谱性质

On the Estrada Index and Spectral Properties of $k$-Uniform Hypergraphs

本文研究$k$-一致超图的谱性质，估计最大特征值之和的界，刻画二特征值超图，建立Estrada指数界并确定单圈超图中第二大Estrada指数。

On the $α$-spectral radius of hypergraphs

For real $α\in [0,1)$ and a hypergraph $G$, the $α$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_α(G)=αD(G)+(1-α)A(G)$, where $A(G)$ is the adjacency matrix of $G$, which is a symmetric matrix with zero diagonal such that for distinct vertices $u,v$ of $G$, the $(u,v)$-entry of $A(G)$ is exactly the number of edges containing both $u$ and $v$, and $D(G)$ is the diagonal matrix of row sums of $A(G)$. We study the $α$-spectral radius of a hypergraph that is uniform or not necessarily uniform. We propose some local grafting operations that increase or decrease the $α$-spectra…

The $(p,q)$-spectral radii of $(r,s)$-directed hypergraphs

An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices in tail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directed hypergraph. For any real numbers $p$, $q\geq 1$, we define the $(p,q)$-spectral radius $λ_{p,q}(G)$ as \[ λ_{p,q}(G):=\max_{||{\bf x}||_p=||{\bf y}||_q=1} \sum_{e\in E(G)}\Bigg(\prod_{u\in T(e)}x_u\Bigg)\Bigg(\prod_{v\in H(e)}y_v\Bigg), \] where ${\bf x}=(x_1, \ldots, x_m)^{\rm T}$, ${\bf y}=(y_1,\ldots, y_n)^{\rm T}$ are real vectors; and $T(e)$, $H(e)$ are the tail and head of arc $e$, respectively. We study some properties about $λ_{p,q}(G)$ …

Monochromatic Edges in Complete Multipartite Hypergraphs

Consider the following problem. In a school with three classes containing $n$ students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occurs when the classes are all boy, all girl and half-and-half. In this paper, we solve many generalizations of the problem including when the school has more than three classes, when triples are replaced by groups of larger sizes, when the classes are of different sizes, and when ge…

Spectra of Uniform Hypergraphs

We present a spectral theory of hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of "hyperdeterminants" of hypermatrices, a.k.a. multidimensional arrays. Hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variati…

超图自同构相容矩阵的谱分解

Spectral decomposition of hypergraph automorphism compatible matrices

本文研究超图自同构与矩阵谱的关系，证明任意自同构相容矩阵的谱可分解为分量旋转对应的小矩阵谱，并将结论推广到单位自同构情形。

A Bound on the Spectral Radius of Hypergraphs with $e$ Edges

For $r\geq 3$, let $f_r\colon [0,\infty)\to [1,\infty)$ be the unique analytic function such that $f_r({k\choose r})={k-1\choose r-1}$ for any $k\geq r-1$. We prove that the spectral radius of an $r$-uniform hypergraph $H$ with $e$ edges is at most $f_r(e)$. The equality holds if and only if $e={k\choose r}$ for some positive integer $k$ and $H$ is the union of a complete $r$-uniform hypergraph $K_k^r$ and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs.