Raffaella Mulas
The graph Cheeger constant and Cheeger inequalities are generalized to the case of hypergraphs whose edges have the same cardinality. In particular, it is shown that the second largest eigenvalue of the generalized normalized Laplacian is bounded both above and below by the generalized Cheeger constant, and the corresponding eigenfunctions can be used to approximate the Cheeger cut.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A Cheeger Inequality for Hypergraphs and Its Applications
本文为非均匀超图建立谱框架，证明 Cheeger 不等式，并构造对 Alon--Boppana 界紧的最优超图扩展器族。
Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the Cheeger constant is generalized and it is shown that the classical Cheeger bounds can be generalized for some classes of hypergraphs; it is shown that a geometric quantity used to study zonotopes bounds the largest eigenvalue from below, and that the notion of coloring number can be generalized and…
We derive Cheeger inequalities for directed graphs and hypergraphs using the reweighted eigenvalue approach that was recently developed for vertex expansion in undirected graphs [OZ22,KLT22,JPV22]. The goal is to develop a new spectral theory for directed graphs and an alternative spectral theory for hypergraphs. The first main result is a Cheeger inequality relating the vertex expansion $\vecψ(G)$ of a directed graph $G$ to the vertex-capacitated maximum reweighted second eigenvalue $\vecλ_2^{v*}$: \[ \vecλ_2^{v*} \lesssim \vecψ(G) \lesssim \sqrt{\vecλ_2^{v*} \cdot \log (Δ/\vecλ_2^{v*})}. \] …
Normalized Laplacian eigenvalues of hypergraphs
本文研究非一致超图的归一化拉普拉斯特征值，给出紧界，并建立边版本交错定理、Cheeger不等式和差异不等式。
Equivalent spectral theory for fundamental graph cut problems
本文提出了一种针对基本图割问题的等价谱理论方法，通过集合对Lovász扩展和Dinkelbach方案，建立了特征对与最优图割问题的一一对应关系，并系统研究了特征向量和特征值的性质。
Spectral theory of dense hypergraph limits
本文发展了超图极限的谱理论，证明了在 $1$-割度量下收敛的超图序列邻接矩阵和 Laplacian 矩阵谱的收敛性，并探讨了相关算子关于不同割范数的连续性。
Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces
本文提出集对函数的Choquet扩张等概念，构建集合分数阶优化的等价表示，推导测度空间的谱界，建立全局扩张常数的单调性不等式，为多类结构的相关几何与组合量提供新的界。
Continuous iterative algorithms for anti-Cheeger cut
本文针对反Cheeger割问题提出一种无需舍入的连续迭代算法（CIA），其子问题具有显式解析解且可在有限步内收敛到局部最优，并能与最大割迭代结合提升解质量，在G-set上实验表明其解质量与启发式方法相当且计算速度更快。
The cut method has been proved to be extremely useful in chemical graph theory. In this paper the cut method is extended to hypergraphs. More precisely, the method is developed for the Wiener index of $k$-uniform partial cube-hypergraphs. The method is applied to cube-hypergraphs and hypertrees. Extensions of the method to hypergraphs arising in chemistry which are not necessary $k$-uniform and/or not necessary linear are also developed.
For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $λ(G) \leq h(G)$, where $λ(G)$ is the second smallest eigenvalue of the Laplacian of a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of $G$. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs). Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generali…
Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes
本研究衔接超图研究的两大分支，分析单纯复形上不同游走的谱隙与超图扩张、电导率的关联，证明部分游走谱隙无法推导超图电导率界，给出类Cheeger不等式，并验证链环扩张无法以类Cheeger方式界超图扩张。
Dual Cheeger Constants, Signless 1-Laplacians and Maxcut
本文针对最大割问题，通过对偶Cheeger常数及其修正形式，提出逆幂算法求解无符号1-Laplacian的谱问题，首次给出G-set上的近似值，并证明递归谱割框架下最坏情况近似比下界不超过0.769。
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on…
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…
In this paper we study fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. To prepare the ground, we define various types of subhypergraphs, as well as various types of walks in a hypergraph. We then prove a number of new results involving cut edges, cut vertices, and blocks. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph.
Let ϕ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, ϕ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if λ_k$ is a constant for some constant k. This provides…
An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size $m/2 + Ω(\sqrt{m})$, and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is $Ω(\sqrt m)$ larger…
In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts re…
We introduce a family of multi-way Cheeger-type constants $\{h_k^σ, k=1,2,\ldots, n\}$ on a signed graph $Γ=(G,σ)$ such that $h_k^σ=0$ if and only if $Γ$ has $k$ balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, those measures of bipartiteness introduced by Desai-Rao, Trevisan, Bauer-Jost, respectively, on unsigned graphs,, and the frustration index (originally called the line index of balance by Harary) on signed graphs. We further unify th…
The min-cut function of weighted hypergraphs and the von Neumann entropy of pure quantum states are both symmetric submodular functions. In this note, we explain this coincidence by proving that the min-cut function of any weighted hypergraph can be approximated (up to an overall rescaling) by the entropies of quantum states known as stabilizer states. This implies that the min-cuts of hypergraphs are constrained by quantum entropy inequalities, and it shows that the recently defined hypergraph cones are contained in the quantum stabilizer entropy cones, as has been conjectured in the recent l…
Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting A_1,...,A_k is the number one would expect to find in a random k-uniform hypergraph. Can we then infer that H is quasi-random? We show that the answer is negative if and only if a_1=...=a_k=1/k. This resolves an open problem raised in 1991 by Chung and Graham [J. AMS '91]. While hypergraphs satisfying the property corresponding to a_1=...=a_k=1/k are not necess…
We consider the Hypergraph-$k$-cut problem. The input consists of a hypergraph $G=(V,E)$ with non-negative hyperedge-costs $c: E\rightarrow R_+$ and a positive integer $k$. The objective is to find a least-cost subset $F\subseteq E$ such that the number of connected components in $G-F$ is at least $k$. An alternative formulation of the objective is to find a partition of $V$ into $k$ non-empty sets $V_1,V_2,\ldots,V_k$ so as to minimize the cost of the hyperedges that cross the partition. Graph-$k$-cut, the special case of Hypergraph-$k$-cut obtained by restricting to graph inputs, has receive…
A sharp higher-order Cheeger inequality
本文证明了归一化拉普拉斯算子的第 k 个特征值与 k 个不相交集合的最大电导之间存在多对数阶的 Cheeger 不等式，并给出了构造性证明，为谱聚类提供了理论保证。
Nearly tight bounds for MaxCut in hypergraphs
该研究给出超图MaxCut近紧界：证明Conlon-Fox-Kwan-Sudakov猜想近似版本（k>k₀时超图r-割超随机割Ω(m^{2/3-ε})），且k≥4或r≥3时线性超图r-割超随机割Ω(m^{3/4})（紧界，证Räty-Tomon猜想）。
We establish that in the large degree limit, the value of certain optimization problems on sparse random hypergraphs is determined by an appropriate Gaussian optimization problem. This approach was initiated in Dembo et. al.(2016) for extremal cuts of graphs. The usefulness of this technique is further illustrated by deriving the optimal value for Max $q$-cut on Erdős-Rényi and random regular graphs, Max XORSAT on Erdős-Rényi hypergraphs, and the min-bisection for the Stochastic Block Model.
The trace and Estrada index of uniform hypergraphs with cut vertices
本文研究含割点的一致超图邻接张量的迹，推导局部扰动下迹与Estrada指数的结果，并证明固定边数超树中超路唯一最小、超星唯一最大。
The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $°^+(v) \le k$ or (indegree) $°^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, int…
An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists α(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{α(i)}$. The Turán number $ex(n,{\cal H})$ of an $r$-graph ${\cal H}$ is the largest size of an $n$-vertex $r$-graph that does not contain ${\cal H}$. A cross-cut of ${\cal H}$ is a set of vertices in ${\cal H}$ that contains exactly one vertex of each edge of ${\cal H}$. The cross-cut number $σ({\cal H})$ of ${\cal H}$ is the minimum size of a cross-cut of ${\cal H}$. We show that for a l…
The trace of uniform hypergraphs with application to Estrada index
本文研究超图邻接张量的迹，给出超树和线性单圈超图的迹的新表达式及割点扰动结果，并据此确定固定边数和完美匹配下Estrada指标最大的唯一超树，以及固定边数和围长3下Estrada指标最大的唯一单圈超图。
Despite the fact that many important problems (including clustering) can be described using hypergraphs, theoretical foundations as well as practical algorithms using hypergraphs are not well developed yet. In this paper, we propose a hypergraph modularity function that generalizes its well established and widely used graph counterpart measure of how clustered a network is. In order to define it properly, we generalize the Chung-Lu model for graphs to hypergraphs. We then provide the theoretical foundations to search for an optimal solution with respect to our hypergraph modularity function. T…
In this note we give a new effective proof method for the equivalence of the notions of testability and nondeterministic testability for uniform hypergraph parameters. We provide the first effective upper bound on the sample complexity of any nondeterministically testable $r$-uniform hypergraph parameter as a function of the sample complexity of its witness parameter for arbitrary $r$. The dependence is of the form of an exponential tower function with the height linear in $r$. Our argument depends crucially on the new upper bounds for the $r$-cut norm of sampled $r$-uniform hypergraphs. We em…
On harmonious coloring of hypergraphs
本文研究超图的和谐着色问题，利用A. Bernshteyn的局部割引理，为最大度为Δ且具有m条边的k-一致超图的和谐数给出了上界h(H)=O(√[k]{k!m})的新证明。
In this paper we study three substructures in hypergraphs that generalize the notion of an Euler tour in a graph. A flag-traversing tour of a hypergraph corresponds to an Euler tour of its incidence graph, hence complete characterization of hypergraphs with an Euler tour follows from Euler's Theorem. An Euler tour is a closed walk that traverses each edge of the hypergraph exactly once; and an Euler family is a family of closed walks that cannot be concatenated and that jointly traverse each edge of the hypergraph exactly once. Lonc and Naroski have shown that the problem of existence of an Eu…
The multiplicity of the zero Laplacian eigenvalue of uniform hypertrees
该研究通过刻画含割点或悬挂边的一致超图的 Laplacian 特征多项式及 Laplacian 匹配多项式，确定一致超树零 Laplacian 特征值的重数，并证明已有文献中的相关猜想。
The replica symmetric solution for hypergraph independent sets in the critical regime
本文证明了一个变分公式，用于描述随机图中避免团等组合问题中非存在概率的对数渐近，该公式在整个临界区域成立，并基于超图硬核模型的副本对称Bethe自由能，通过窗口熵方法和信念传播引导的贪婪算法证明。
Re-imagining Spectral Graph Theory
提出内积拉普拉斯算子，统一组合图、归一化图、超图及有向图的拉普拉斯算子，推导广义等周不等式，恢复Dirichlet和Neumann子图特征值，为现有拉普拉斯算子提供新研究语境并增强应用灵活性。
Distributional limits of graph cuts on discretized grids
本文研究了图割在离散网格上的分布极限，揭示了不同图割方法的极限分布特性，并通过实验验证了理论结果在实际应用中的有效性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。