Despite of the extreme success of the spectral graph theory, there are relatively few papers applying spectral analysis to hypergraphs. Chung first introduced Laplacians for regular hypergraphs and showed some useful applications. Other researchers treated hypergraphs as weighted graphs and then studied the Laplacians of the corresponding weighted graphs. In this paper, we aim to unify these very different versions of Laplacians for hypergraphs. We introduce a set of Laplacians for hypergraphs through studying high-ordered random walks on hypergraphs. We prove the eigenvalues of these Laplacians can effectively control the mixing rate of high-ordered random walks, the generalized distances/diameters, and the edge expansions.
We generalize the notions of Laplacian and signless Laplacian Estrada index to uniform hypergraphs. For an $r$-uniform hypergraph $H,$ we derive an order $r+1$ trace formula of the (signless) Laplacian tensor of $H.$ Among others by using this trace formula, we obtain lower bounds for the signless Laplacian Estrada index and upper bounds for the Laplacian Estrada index. Moreover, we establish a bound involving both the Laplacian Estrada index and Laplacian energy of a uniform hypergraph.
An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate tha…
Let $\mathcal{H}$ be a connected $k$-uniform hypergraph on $n$ vertices and $m$ hyperedges. In [A.~Banerjee, On the spectrum of hypergraph, Linear Algebra and its Application, 614(2021), 82--110], Anirban Banerjee introduced a new adjacency matrix for hypergraphs. In this article we consider the corresponding signless Laplacian matrix $Q(\mathcal{H})$ and discuss about its spectrum.
The graph entropy describes the structural information of graph. Motivated by the definition of graph entropy in general graphs, the graph entropy of hypergraphs based on Laplacian degree are defined. Some results on graph entropy of simple graphs are extended to k-uniform hypergraphs. Using an edge-moving operation, the maximum and minimum graph entropy based on Laplacian degrees are determined in k-uniform hypertrees, unicyclic k-uniform hypergraphs, bicyclic k-uniform hypergraphs and k-uniform chemical hypertrees, respectively, and the corresponding extremal graphs are determined.
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.
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…
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 signle…
Let $\mathcal{A(}G\mathcal{)},\mathcal{L(}G\mathcal{)}$ and $\mathcal{Q(}% G\mathcal{)}$ be the adjacency tensor, Laplacian tensor and signless Laplacian tensor of uniform hypergraph $G$, respectively. Denote by $λ(\mathcal{T})$ the largest H-eigenvalue of tensor $\mathcal{T}$. Let $H$ be a uniform hypergraph, and $H^{\prime}$ be obtained from $H$ by inserting a new vertex with degree one in each edge. We prove that $λ(\mathcal{Q(}% H^{\prime}\mathcal{)})\leqλ(\mathcal{Q(}H\mathcal{)}).$ Denote by $G^{k}$ the $k$th power hypergraph of an ordinary graph $G$ with maximum degree $Δ\geq2$. We will…
The $p$-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex $p$-Laplacian and a hyperedge $p$-Laplacian are defined for oriented hypergraphs, for all $p\geq 1$. Several spectral properties of these operators are investigated.
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…
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 att…
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.
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…
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.
The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a ratio--like bound are shown. A Sandwich Theorem involving the clique number, the vector chromatic number and the coloring number is proved, as well as a lower bound for the vector chromatic number in terms of the smallest and the largest eigenvalue of the normalized Laplacian. In addition, spectral partition numbers are studied in relation to the coloring numb…
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.
In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2Δ(G)$, where $Δ(G)$ is the largest degree of $G$, if and only if $G$ is regular. Thus the largest Laplacian H-eigenvalue of $G$, reaches the same upper bound, if and only if $G$ is regular and odd-bipartite. We show that an $s$-cycle $G$, as a $k$-uniform hypergraph, where $1 \le s \le k-1$, is regular if and only if there is a positive integer $q$ such that $k=q(k-s)$. We show that an even-uniform $s$-path and an even-uniform non-regular…
In this paper, we introduce the class of cored hypergraphs and power hypergraphs, and investigate the properties of their Laplacian H-eigenvalues. From an ordinary graph, one may generate a $k$-uniform hypergraph, called the $k$th power hypergraph of that graph. Power hypergraphs are cored hypergraphs, but not vice versa. Hyperstars, hypercycles, hyperpaths are special cases of power hypergraphs, while sunflowers are a subclass of cored hypergraphs, but not power graphs in general. We show that the largest Laplacian H-eigenvalue of an even-uniform cored hypergraph is equal to its largest signl…
Norwegian University of Science and Technology（挪威科技大学）；BIFOLD—Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）；Technical University Berlin（柏林工业大学）；Zuse Institute Berlin（齐泽柏林研究所）
We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph.
Weighted hypergraphs are generalizations of weighted simplicial complexes. In recent years, weighted Laplacians of weighted simplicial complexes have been studied. In 2016, as a generalization of the homology of simplicial complexes, the embedded homology of hypergraphs was constructed. In this paper, we generalize the weighted Laplacians of weighted simplicial complexes to weighted hypergraphs. We study the relations between the weighted Laplacians and the weighted embedded homology of weighted hypergraphs. We generalize the Hodge decompositions of weighted simplicial complexes to weighted hy…
In this note we provide a combinatorial interpretation for the powers of the hypergraph Laplacians. Our motivation comes from the discrete formulation of quantum mechanics and thermodynamics in the case of finite graphs, which suggest a natural extension to simplicial and CW-complexes. With this motivation, we also define generalizations of the odd Laplacian which is specific to hypergraphs arising from CW-complexes. We then provide a combinatorial interpretation for the powers of these Laplacians.
In this paper, we present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.
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 t…
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 grap…
Hypergraphs are a generalization of graphs in which edges can connect any number of vertices. They allow the modeling of complex networks with higher-order interactions, and their spectral theory studies the qualitative properties that can be inferred from the spectrum, i.e. the multiset of the eigenvalues, of an operator associated to a hypergraph. It is expected that a small perturbation of a hypergraph, such as the removal of a few vertices or edges, does not lead to a major change of the eigenvalues. In particular, it is expected that the eigenvalues of the original hypergraph interlace th…
An Euler tour in a hypergraph $H$ is a closed walk that traverses each edge of $H$ exactly once, and an Euler family is a family of closed walks that jointly traverse each edge of $H$ exactly once. An $\ell$-covering $k$-hypergraph, for $2 \le \ell < k$, is a $k$-uniform hypergraph in which every $\ell$-subset of vertices lie together in at least one edge. In this paper we prove that every $\ell$-covering $k$-hypergraph, for $k \ge 3$, admits an Euler family.
In this paper, we study energies associated with hypergraphs. More precisely, we obtain results for the incidence and the singless Laplacian energies of uniform hypergraphs. In particular, we obtain bounds for the incidence energy as functions of well known parameters, such as maximum degree, Zagreb index and spectral radius. We also relate the incidence and signless Laplacian energies of a hypergraph with the adjacency energies of its subdivision graph and line multigraph, respectively. In addition, we compute the signless Laplacian energy for the class of the power hypergraphs.
Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different structural properties of a hypergrpah, can be well studied using spectral properties of these matrices. Connectivity of a hypergraph is also investigated by the eigenvalues of these operators. Spectral radii of the same are bounded by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by the eigenvalues of its connectivity matrices. We charac…
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 …
An Euler tour of a hypergraph is a closed walk that traverses every edge exactly once; if a hypergraph admits such a walk, then it is called eulerian. Although this notion is one of the progenitors of graph theory --- dating back to the eighteenth century --- treatment of this subject has only begun on hypergraphs in the last decade. Other authors have produced results about rank-2 universal cycles and 1-overlap cycles, which are equivalent to our definition of Euler tours. In contrast, an Euler family is a collection of nontrivial closed walks that jointly traverse every edge of the hypergrap…
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 si…
For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are investigated including the incidence dual, the intersection graph (line graph), and the 2-section. A connection is then made between oriented hypergraphs and balanced incomplete block designs.
The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries…
The spectral theory of higher-order symmetric tensors is an important tool to reveal some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop a first-order optimization algorithm for computing H- and Z-eigenvalues of these large scale sparse tensors (CEST). With the aid of the Kurdyka-Łojasiewicz property, we prove that the sequence of iterates generated…
Let $\mathcal{H}$ be a uniform hypergraph. Let $\mathcal{A(H)}$ and $\mathcal{Q(H)}$ be the adjacency tensor and the signless Laplacian tensor of $\mathcal{H}$, respectively. In this note we prove several bounds for the spectral radius of $\mathcal{A(H)}$ and $\mathcal{Q(H)}$ in terms of the degrees of vertices of $\mathcal{H}.$
Let $H=(V,E)$ be an $r$-uniform hypergraph with the vertex set $V$ and the edge set $E$. For $1\leq s \leq r/2$, we define a weighted graph $G^{(s)}$ on the vertex set ${V\choose s}$ as follows. Every pair of $s$-sets $I$ and $J$ is associated with a weight $w(I,J)$, which is the number of edges in $H$ passing through $I$ and $J$ if $I\cap J=\emptyset$, and 0 if $I\cap J\not=\emptyset$. The $s$-th Laplacian $Ł^{(s)}$ of $H$ is defined to be the normalized Laplacian of $G^{(s)}$. The eigenvalues of $\mathcal L^{(s)}$ are listed as $λ^{(s)}_0, λ^{(s)}_1,..., λ^{(s)}_{{n\choose s}-1}$ in non-decr…
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information.
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, $\Q(G^{k,\frac{k}{2}})$) con…
Let $\mathbb{Q}_{k,n}$ be the set of the connected $k$-uniform weighted hypergraphs with $n$ vertices, where $k,n\geq 3$. For a hypergraph $G\in \mathbb{Q}_{k,n}$, let $\mathcal{A}(G)$, $\mathcal{L} (G)$ and $\mathcal{Q} (G)$ be its adjacency tensor, Laplacian tensor and signless Laplacian tensor, respectively. The spectral radii of $\mathcal{A}(G)$ and $\mathcal{Q} (G)$ are investigated. Some basic properties of the $H$-eigenvalue, the $H^{+}$-eigenvalue and the $H^{++}$-eigenvalue of $\mathcal{A}(G)$, $\mathcal{L} (G)$ and $\mathcal{Q} (G)$ are presented. Several lower and upper bounds of th…
