Reinhard Diestel
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with fundamental cuts and cycles playing their usual roles.
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We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its zero-th and first homology groups with coefficients in $\mathbb{F}_2$ vanish. Although this is not intrinsically a monotone property, we show that it nevertheless has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance of the last minimal obstruction.
An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families -- balanced, balanceable, and unbalanceable -- and we obtain a complete classification of the balanced circuits of oriented 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 …
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…
An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camion's algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterizati…
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…
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 $p$-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.
In this paper we introduce the definition of transitivity for oriented 3-hypergraphs in order to study partial and complete cyclic orders. This definition allow us to give sufficient conditions on a partial cyclic order to be totally extendable. Furthermore, we introduce the 3-hypergraph associated to a cyclic permutation and characterize it in terms of cyclic comparability 3-hypergraphs.
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 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 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 oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.
Let $G$ be an $r$-uniform hypergraph. When is it possible to orient the edges of $G$ in such a way that every $p$-set of vertices has some $p$-degree equal to $0$? (The $p$-degrees generalise for sets of vertices what in-degree and out-degree are for single vertices in directed graphs.) Caro and Hansberg asked if the obvious Hall-type necessary condition is also sufficient. Our main aim is to show that this is true for $r$ large (for given $p$), but false in general. Our counterexample is based on a new technique in sparse Ramsey theory that may be of independent interest.
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…
Beginning with the concepts of orientation for a 3-hypergraph and transitivity for an oriented 3-hypergraph, it is natural to study the class of comparability 3-hypergraphs (those that can be transitively oriented). In this work we show three different behaviors in respect to the relationship between the chromatic number and the clique number of a comparability 3-hypergraph, this is in contrast with the fact that a comparability simple graph is a perfect graph.
Homological invariants of edge ideals of weighted oriented graphs
该研究确定了固定顶点数加权有向图边理想的深度、维数和正则性三元组，并计算了固定顶点数加权有向树及二部图边理想的贝蒂表大小。
Oriented hypergraphs and generalizing the Harary-Sachs theorem to integer matrices
本文通过有向超图上的贡献者概念推广 Harary-Sachs 定理，刻画整数矩阵 Laplacian 的全子式多项式系数，并证明尾等价分组的相消性及 Laplacian 与其对偶的非零等谱性。
Chemically inspired Erdős-Rényi oriented hypergraphs
本文提出有向超图的 Erdős-Rényi 随机模型，揭示大顶点数下有向超边期望数与期望度或大小之比为 3/2，并论证其在建模大规模化学反应及化学演化分析中的适用性。
Erdős [On Schütte problem, Math. Gaz. 47 (1963)] proved that every tournament on $n$ vertices has a directed dominating set of at most $\log (n+1)$ vertices, where $\log$ is the logarithm to base $2$. He also showed that there is a tournament on $n$ vertices with no directed domination set of cardinality less than $\log n - 2 \log \log n + 1$. This notion of directed domination number has been generalized to arbitrary graphs by Caro and Henning in [Directed domination in oriented graphs, Discrete Appl. Math. (2012) 160:7--8.]. However, the generalization to directed r-uniform hypergraphs seems…
Hypergraphic zonotopes and acyclohedra
本文提出超图带状体及其特例无环多面体，将其体积与超树计数关联，并连接无环定向概念，推广了图形带状体和置换多面体理论。
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…
Caro, West, and Yuster studied how $r$-uniform hypergraphs can be oriented in such a way that (generalizations of) indegree and outdegree are as close to each other as can be hoped. They conjectured an existence result of such orientations for sparse hypergraphs, of which we present a proof.
In this paper we derive and present an application of hypergraphic oriented matroids for the purpose of enumerating the variable interdependencies that define the chemical complexes associated with the kinetics of non-linear dynamical system representations of chemical kinetic reaction flow networks. The derivation of a hypergraphic oriented matroid is obtained by defining a closure operator on families of n-subsets of signed multi-sets from which a "Z-module" is obtained. It has been observed that every instantiation of the closure operator on the signed multiset families define a matroid str…
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobject classifier of incidence hypergraphs. Moreover, the injective envelope is calculated and shown to contain the class of uniform hypergraphs -- providing a combinatorial framework for the entries of incidence matrices. A multivariable all-minors characteristic polynomial is obtained for both the determinant and permanent of the oriented hypergraphic…
Source characterization of the hypergraphic posets
本文将 Bergeron 和 Pilaud 对区间超图偏序集的源刻画推广到任意超图，建立了超图偏序集与无环定向源之间的对应关系。
The notions of spectral measures and spectral classes, which are well known for graphs, are generalized and investigated for oriented hypergraphs.
$p$-Laplacian Operators on Hypergraphs
本文推广了有向图与有向超图上的微分算子，提出面向顶点和（超）弧的梯度、伴随算子及 $p$-拉普拉斯算子的新参数化定义，使其满足连续情形的预期性质。
An oriented $k$-uniform hypergraph is said to have Property O if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. Recently Duffus, Kay and Rödl investigated the minimum number $f(k)$ of edges in a $k$-uniform hypergaph with Property O. They proved that $k! \leq f(k) \leq (k^2 \ln k) k!$, where the upper bound holds for $k$ sufficiently large. In this short note we improve their upper bound by a factor of $k \ln k$, showing that $f(k) \le \left(\lfloor \frac{k}{2} \rfloor +1 \right) k! - \lfloor \frac{k}{2} \rfloor (k-1)!$ for every $k\ge…
Property O and Erdős--Szekeres properties in linear hypergraphs
本文研究线性超图中的性质 O，确定最小边数至多差多项式因子，并给出顶点数界及 Erdős–Szekeres 性质下的最小规模。
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。
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to these questions (which is not quite the predicted one). Our approach is based on the recent `hypergraph containers' method, developed independently by Saxton and Thomason as well as by Balogh, Morris and Samotij. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. Along…
Simplicial Complex Emergence on Directed Hypergraphs
研究有向超图上高阶网络的单纯复形涌现条件，通过张量分解和范数分析提出稳定性判定方法，并展示同调工具在适应性系统中的应用
The Topological Structures of the Orders of Hypergraphs
该研究从范畴论角度探索并完成了二元关系的三个基本视角（超图关联矩阵、概念格形式背景、单纯复形上的拓扑余层）之间的关系映射，提出整合性函子框架并给出三个新结果，通过例子说明概念及与以往结果的联系。
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