In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a k-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval [0, 2], and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H+-eigenvalues of the Laplacian and all the smallest H+-eigenvalues of its sub-t...

In this work we propose a model to represent the web as a directed hypergraph (instead of a graph), where links connect pairs of disjointed sets of pages. The web hypergraph is derived from the web graph by dividing the set of pages into non-overlapping blocks and using the links between pages of distinct blocks to create hyperarcs. A hyperarc connects a block of pages to a single page, in orde...

In a visual cryptography scheme, a secret image is encoded into n shares, in the form of transparencies. The shares are then distributed to n participants. Qualified subsets of participants can recover the secret image by superimposing their transparencies, but non-qualified subsets of participants have no information about the secret image. Pixel expansion, which represents the number of subpi...

Given a square-free monomial ideal $I$ in polynomial ring $R$ over field $\mathbb{K}$, one can associate it with its LCM-lattice and hypergraph. In this short note, we establish the connection between hypergraph, doing so provide sufficient condition for removing higher dimension edges of hypergraph without impacting projective ideal. We also offer algorithms to compute class ideals built using...

Generalizing the concept of dense hypergraph, we say that a hypergraph with n edges is weakly dense, if no k in half-open interval [2,n) degree more than k2 vertices. In our main result, prove famous Erdős–Faber–Lovász conjecture when dense.

Abstract Hypergraph computation has attracted much attention and shown apparent advantages in many application fields, such as computer vision, social networks, biomedicine. In this book, we systematically introduce the basic knowledge, algorithms, applications of hypergraph three parts discuss some recent progress direction.

One of the earliest results in Combinatorics is Mantel’s theorem from 1907 that the largest triangle-free graph on a given vertex set is complete bipartite. However, a seemingly similar question posed by Turán in 1941 is still open: what is the largest 3-uniform hypergraph on a given vertex set with no tetrahedron? This question can be considered a test case for the general hypergraph Turán pro...

A hypergraph is a set V of vertices and a set E of non-empty subsets of V , called hyperedges. Unlike graphs, hypergraphs can perform higher-order interactions in social and communication networks. Directed hypergraphs are much like directed graphs. Colors are used to distinguish the classes. Coloring a hypergraph H must assign atleast two different colors to the vertices of every hyperedge. Th...

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. The research on this model for graphs has been started by Bohman et al. in [7], and continued in [8] and [16]. Here we obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergr...

Graph partitioning is an important and well studied problem in combinatorial scientific computing, and is commonly used to reduce communication in parallel computing. Different models (graph, hypergraph) and objectives (edge cut, boundary vertices) have been proposed. Hypergraph partitioning has become increasingly popular over the last decade. Its main strength is that it accurately captures c...