Class Of Hypergraphs Research Articles

Some results and problems on clique coverings of hypergraphs

AbstractFor a ‐uniform hypergraph we consider the parameter , the minimum size of a clique cover of the edge set of . We derive bounds on for belonging to various classes of hypergraphs.

On small non-uniform hypergraphs without property B

For a given hypergraph H = (V, E) consider the sum q(H) of 2^{−|e|} over e ∈ E. Consider the class of hypergraphs whose smallest edge is of size n and for which every 2-colouring has a monochromatic edge. Let q(n) be the smallest value of q(H) in this class. We provide a survey of the known bounds on q(n) and make some minor refinements.

On some general operators of hypergraphs

Here we introduce connectivity operators, namely, diffusion operators, general Laplacian operators, and general adjacency operators for hypergraphs. These operators are generalisations of some conventional notions of apparently different connectivity matrices associated with hypergraphs. In fact, we introduce here a unified framework for studying different variations of the connectivity operators associated with hypergraphs at the same time. Eigenvalues and corresponding eigenspaces of the general connectivity operators associated with some classes of hypergraphs are computed. Applications such as random walks on hypergraphs, dynamical networks, and disease transmission on hypergraphs are studied from the perspective of our newly introduced operators. We also derive spectral bounds for the weak connectivity number, degree of vertices, maximum cut, bipartition width, and isoperimetric constant of hypergraphs.

Stability theorems for some Kruskal–Katona type results

The classical Kruskal–Katona theorem gives a tight upper bound for the size of an r-uniform hypergraph H as a function of the size of its shadow. Its stability version was obtained by Keevash who proved that if the size of H is close to the maximum with respect to the size of its shadow, then H is structurally close to a complete r-uniform hypergraph. We prove similar stability results for two classes of hypergraphs whose extremal properties have been investigated by many researchers: the cancellative hypergraphs and hypergraphs without expansion of cliques.

Six methods for transforming layered hypergraphs to apply layered graph layout algorithms

AbstractHypergraphs are a generalization of graphs in which edges (hyperedges) can connect more than two vertices—as opposed to ordinary graphs where edges involve only two vertices. Hypergraphs are a fairly common data structure but there is little consensus on how to visualize them. To optimize a hypergraph drawing for readability, we need a layout algorithm. Common graph layout algorithms only consider ordinary graphs and do not take hyperedges into account. We focus on layered hypergraphs, a particular class of hypergraphs that, like layered graphs, assigns every vertex to a layer, and the vertices in a layer are drawn aligned on a linear axis with the axes arranged in parallel. In this paper, we propose a general method to apply layered graph layout algorithms to layered hypergraphs. We introduce six different transformations for layered hypergraphs. The choice of transformation affects the subsequent graph layout algorithm in terms of computational performance and readability of the results. Thus, we perform a comparative evaluation of these transformations in terms of number of crossings, edge length, and impact on performance. We also provide two case studies showing how our transformations can be applied to real‐life use cases. A copy of this paper with all appendices and supplemental material is available at osf.io/grvwu.

Geometric vs algebraic nullity for hyperpaths

We consider the question of how the eigenvarieties of a hypergraph relate to the algebraic multiplicities of their corresponding eigenvalues. Specifically, we (1) fully describe the irreducible components of the zero-eigenvariety of a loose $3$-hyperpath (its nullvariety), (2) use recent results of Bao-Fan-Wang-Zhu to compute the corresponding algebraic multiplicity of zero (its nullity), and then (3) for this special class of hypergraphs, verify a conjecture of Hu-Ye about the relationship between the geometric (multi-)dimension of the nullvariety and the nullity.

Optimal Communication Rates and Combinatorial Properties for Common Randomness Generation

We study common randomness generation problems where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> players aim to generate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">same</i> sequences of random coin flips where some subsets of the players share an independent common coin which can be tossed multiple times, and there is a publicly seen blackboard through which the players communicate with each other. We provide a tight representation of the optimal communication rates via linear programming, and more importantly, propose explicit algorithms for the optimal distributed simulation for a wide class of hypergraphs. In particular, the optimal communication rate in complete hypergraphs is still achievable in sparser hypergraphs containing a path-connected cycle-free cluster of topologically connected components. Some key steps in analyzing the upper bounds rely on two different definitions of connectivity in hypergraphs, which may be of independent interest.

Sharp Bounds on the Spectral Radii of Uniform Hypergraphs concerning Diameter or Clique Number

In this paper, we defined two classes of hypergraphs, hyperbugs and kite hypergraphs. We show that balanced hyperbugs maximize the spectral radii of hypergraphs with fixed number of vertices and diameter and kite hypergraphs minimize the spectral radii of hypergraphs with fixed number of vertices and clique number.

All-terminal hypernetwork reliability synthesis of a kind of semi-deterministic hypergraphs

Network reliability plays an important role in analysis, synthesis and detection of real-world networks. In this paper, we first propose the concept of hypernetwork reliability, which generalizes the concept of network reliability. The model for hypernetwork reliability studies consists of a hypergraph with perfect reliable vertices and equal and independent hyperedge failure probability [Formula: see text]. The measure of reliability is defined as the probability that a hypergraph is connected. Let [Formula: see text] be an [Formula: see text]-uniform hypergraph with the number of vertices [Formula: see text] and the number of hyperedges [Formula: see text], where every hyperedge connects [Formula: see text] vertices. We confirm the possibility of the existence of a fixed hypergraph that is optimal or least for all hyperedges same survival possible [Formula: see text]. It is simple to verify that such hypergraph exists if [Formula: see text]. For a kind of 2-regular 3-uniform hypergraphs, we calculate the upper and lower bounds on the all-terminal reliability, and describe the class of hypergraphs that reach the boundary.

Spectral theory of Laplace operators on oriented hypergraphs

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 used for proving a Hoffman-like bound. Finally, the spectrum of the unnormalized Laplacian for Cartesian products of hypergraphs is discussed.

Conditions for a Bigraph to be Super-Cyclic

A hypergraph $\mathcal H$ is super-pancyclic if for each $A \subseteq V(\mathcal H)$ with $|A| \geqslant 3$, $\mathcal H$ contains a Berge cycle with base vertex set $A$. We present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph $\mathcal H$ with $ \delta(\mathcal H)\geqslant \max\{|V(\mathcal H)|, \frac{|E(\mathcal H)|+10}{4}\}$.&#x0D; We also consider super-cyclic bipartite graphs: those are $(X,Y)$-bigraphs $G$ such that for each $A \subseteq X$ with $|A| \geqslant 3$, $G$ has a cycle $C_A$ such that $V(C_A)\cap X=A$. Such graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs.

Generalized hypergraph coloring

A smooth hypergraph property $\mathcal{P}$ is a class of hypergraphs that is hereditary and non-trivial, i.e., closed under induced subhypergraphs and it contains a non-empty hypergraph but not all hypergraphs. In this paper we examine $\mathcal{P}$-colorings of hypergraphs with smooth hypergraph properties $\mathcal{P}$. A $\mathcal{P}$-coloring of a hypergraph $H$ with color set $C$ is a function $\varphi:V(H) \to C$ such that $H[\varphi^{-1}(c)]$ belongs to $\mathcal{P}$ for all $c \in C$. Let $L: V(H) \to 2^C$ be a so called list-assignment of the hypergraph $H$. Then, a $(\mathcal{P},L)$-coloring of $H$ is a $\mathcal{P}$-coloring $\varphi$ of $H$ such that $\varphi(v) \in L(v)$ for all $v \in V(H)$. The aim of this paper is a characterization of $(\mathcal{P},L)$-critical hypergraphs. Those are hypergraphs $H$ such $H-v$ is $(\mathcal{P},L)$-colorable for all $v \in V(H)$ but $H$ itself is not. Our main theorem is a Gallai-type result for critical hypergraphs, which implies a Brooks-type result for $(\mathcal{P},L)$-colorable hypergraphs. In the last section, we prove a Gallai bound for the degree sum of $(\mathcal{P},L)$-critical locally linear hypergraphs.

Ollivier Ricci curvature of directed hypergraphs

Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propose a general definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier’s definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. While the definition looks somewhat complex, in the end we shall be able to express our curvature in a very simple formula, kappa =mu _0-mu _2-2mu _3. This formula simply counts the fraction of vertices that have to be moved by distances 0, 2 or 3 in an optimal transport plan. We can then characterize various classes of hypergraphs by their curvature.

Least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices

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-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.

The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs

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 attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite hypergraphs.

Extremal problems in hypergraph colourings

Extremal problems in hypergraph colouring originate implicitly from Hilbert’s theorem on monochromatic affine cubes (1892) and van der Waerden’s theorem on monochromatic arithmetic progressions (1927). Later, with the advent and elaboration of Ramsey theory, the variety of problems related to colouring of explicitly specified hypergraphs widened rapidly. However, a systematic study of extremal problems on hypergraph colouring was initiated only in the works of Erdős and Hajnal in the 1960s. This paper is devoted to problems of finding edge-minimum hypergraphs belonging to particular classes of hypergraphs, variations of these problems, and their applications. The central problem of this kind is the Erdős–Hajnal problem of finding the minimum number of edges in an -uniform hypergraph with chromatic number at least three. The main purpose of this survey is to spotlight the progress in this area over the last several years. Bibliography: 168 titles.

The Wiener index of hypergraphs

The Wiener index is defined to be the sum of distances between every unordered pair of vertices in a connected hypergraph. In this paper, we first study how the Wiener index of a hypergraph changes under some graft transformations. For $$1\le m\le n-1$$, we obtain the unique hypertree that achieves the minimum (or maximum) Wiener index in the class of hypertrees on n vertices and m edges. Then we characterize the unique hypertrees on n vertices with first three smallest Wiener indices, and the unique hypertree (not 2-uniform) with maximum Wiener index, respectively. In addition, we determine the unique hypergraph that achieves the minimum Wiener index in the class of hypergraphs on n vertices and p pendant edges.

Sprague-Grundy function of matroids and related hypergraphs

We consider a generalization of the classical game of Nim called hypergraph Nim. Given a hypergraph H on the ground set V={1,…,n} of n piles of stones, two players alternate in choosing a hyperedge H∈H and strictly decreasing all piles i∈H. The player who makes the last move is the winner. In this paper we give an explicit formula that describes the Sprague-Grundy function of hypergraph Nim for several classes of hypergraphs. In particular we characterize all 2-uniform hypergraphs (that is graphs) and all matroids for which the formula works. We show that all self-dual matroids are included in this class.

Turán numbers for Berge-hypergraphs and related extremal problems

Let F be a graph. We say that a hypergraph H is a Berge-F if there is a bijection f:E(F)→E(H) such that e⊆f(e) for every e∈E(F). Note that Berge-F actually denotes a class of hypergraphs. The maximum number of edges in an n-vertex r-graph with no subhypergraph isomorphic to any Berge-F is denoted exr(n,Berge-F). In this paper, we investigate the case when F=Ks,t and establish an upper-bound when r≥3, and a lower-bound when r=4 and t is large enough compared to s. Additionally, we prove a counting result for r-graphs of girth five that complements the asymptotic formula ex3(n,Berge-{C2,C3,C4})=16n3∕2+o(n3∕2) of Lazebnik and Verstraëte (2003).

Stability Results for Two Classes of Hypergraphs

Mubayi and Pikhurko established several Turan-type results and stability results for $r$-uniform hypergraphs. In particular, they considered hypergraphs that avoid a copy of an expanded complete 2-...