On the Laplacian Spectrum and Walk-regular Hypergraphs

We use the generalization of the Laplacian matrix to hypergraphs to obtain several spectral-like results on hypergraphs. For instance, we obtain upper bounds on the eccentricity and the excess of any vertex of hypergraphs. We extend to the case of hypergraphs the concepts of walk regularity and spectral regularity, showing that all walk-regular hypergraphs are spectrally-regular. Finally, we obtain an upper bound on the mean distance of walk-regular hypergraphs that involves all the Laplacian spectrum.     

On the Laplacian Spectrum and Walk-regular Hypergraphs

We use the generalization of the Laplacian matrix to hypergraphs to obtain several spectral-like results on hypergraphs. For instance, we obtain upper bounds on the eccentricity and the excess of any vertex of hypergraphs. We extend to the case of hypergraphs the concepts of walk regularity and spectral regularity, showing that all walk-regular hypergraphs are spectrally-regular. Finally, we obtain an upper bound on the mean distance of walk-regular hypergraphs that involves all the Laplacian spectrum.     

秩为r的本原超图的指数集(英文)

本文研究了超图的本原性质.运用图论方法,得到了具有秩r(≥3)的所有n阶本原有向超图的指数集,并刻划了其极超图.     

Higher Dimensional Aztec Diamonds and a (2d + 2)-Vertex Model

Motivated by the close relationship between the number of perfect matchings of the Aztec diamond graph introduced in [5] and the free energy of the square-ice model, we consider a higher dimensional analog of this phenomenon. For d 1, we construct d-uniform hypergraphs which generalize the Aztec diamonds and we consider a companion d-dimensional statistical model (called the 2d + 2-vertex model) whose free energy is given by the logarithm of the number of perfect matchings of our hypergraphs. We prove that the limit defining the free energy per site of the 2d + 2-vertex model exists and we obtain bounds for it. As a consequence, we obtain an especially good asymptotical approximation for the number of matchings of our hypergraphs.     

Applications of hypergraph coloring to coloring graphs not inducing certain trees

We present a simple result on coloring hypergraphs and use it to obtain bounds on the chromatic number of graphs which do not induce certain trees.     

On the distribution of the leading digit of $$a^n$$: a study via $$\chi ^2$$χ2 statistics

A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the location-domination number of hypergraphs in general and exact values with specified conditions are investigated. Moreover, locating-dominating sets in some specific hypergraphs are found.     

On the Number of Independent Sets in Simple Hypergraphs

Extremal problems on the number of j-independent sets in uniform simple hypergraphs are studied. Nearly optimal results on the maximum number of independent sets for the class of simple regular hypergraphs and on the minimum number of independent sets for the class of simple hypergraphs with given average degree of vertices are obtained.     

关于D-完全一致混合超图上色数的一个结论的推广

混合超图的上、下色数的研究是超图研究中一个重要的话题.由于超图本身结构上的复杂性,近年来对超图色性的研究也近局限于对一些特殊图类的研究,其中完全一致混合超图是最为热门的图类之一.给出了D完全(C不完全)一致混合超图的概念,并运用组合数学中有关分划的思想和方法对该图类的色性进行了进一步的研究,对相关文献中给出的结论进行了推广,得到了一个较为一般化的结论.并在该定理的证明中得到并证明了一个关于混合超图C稳定集的重要论断,对超图色性研究有着重要的意义.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

On partitioning of hypergraphs

We study edge-isoperimetric problems (EIP) for hypergraphs and extend some technique in this area from graphs to hypergraphs. In particular, we establish some new results on a relationship between the EIP and some extremal poset problems, and apply them to obtain an exact solution of the EIP for certain hypergraph families. We also show how to solve the EIP on hypergraphs in some cases when the link to posets does not work. Another outcome of our results is a new series of hypergraphs admitting nested solutions in the EIP.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

反超图的笛卡儿积的上色数

讨论反超图的笛卡儿积的着色理论 ,求出了满足一定条件的反超图的笛卡儿积的上色数 .     

Critical hypergraphs and intersecting set-pair systems

A method is given for solving extremal problems of the following type: determine the maximal number of vertices in a class of hypergraphs. Results are applied for τ-critical and ν-critical hypergraphs.     

Paths and cycles of hypergraphs

Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.     

D-完全一致混合超图不可着色的一个充要条件

混合超图的上,下色数与C-超边和D-超边数有着必然联系.一般地,增加C边会使下色数x(H)增加,增加D-超边会使上色数(x)(H)减小.本论文对D-完全一致混合超图进行研究,利用组合数学中分划思想及方法得到的D-完全一致混合超图不可着色的一个充要条件,对D-完全一致混合超图能否着色找到了可行的依据,进一步揭示C-超边数...     

On Extremal Hypergraphs for Hamiltonian Cycles

We study sufficient conditions for Hamiltonian cycles in hypergraphs and obtain both Turán- and Dirac-type results. While the Turán-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields just a sufficient condition relying solely on the minimum vertex degree.     

AN INVARIANT FOR HYPERGRAPHS

ＡＮＩＮＶＡＲＩＡＮＴＦＯＲＨＹＰＥＲＧＲＡＰＨＳＷＡＮＧＪＩＡＮＦＡＮＧ（ＩｎｓｔｉｔｕｔｅｏｆＡＰＰｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，Ｂｅｉｊｉｎｇ１０００８０，ＣｈｉｎａａｎｄＡｓｉａ－Ｐａｃｉｆ...     

On extremal hypergraphs for Hamiltonian cycles

We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Turán- and Dirac-type results. While the Turán-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields a sufficient condition relying solely on the minimum vertex degree.     

On the cyclomatic number of a hypergraph

This note generalizes the notion of cyclomatic number (or cycle rank) from Graph Theory to Hypergraph Theory and links it up with the concept of planarity in hypergraphs which was recently introduced by R.P. Jones. Sharp bounds are obtained for the cyclomatic number of the planar hypergraphs and, further, it is shown that the upper bound is attainable if, and only if the hypergraph satisfies Krewera's condition.     

超图的强星色数

图的星色数的概念是由Ａ．Ｖｉｎｃｅ（１９８８）首次提出来的,它是图的色数的一个自然而又重要的推广,Ｌ．Ｈａｄａｄ等人（１９９４）将这一概念推广到一致超图,定义了ｈ－一致超图的强（弱）星色数,这里我们给出一般超图的强星色数的概念,研究了它的基本性质,计算了３－一致循环超图的强星色数,它们的强星色数形成了一个严格介于３和４之间的递减序列．