A hypergraph G = (X,W ) is called d-uniform if each hyperedge w is a set of d vertices. A 1-factor of a hypergraph G is a set of hyperedges such that every vertex of the hypergraph is incident to exactly one hyperedge from the set. A 1factorization of G is a partition of all hyperedges of the hypergraph into disjoint 1-factors. The adjacency matrix of a d-uniform hypergraph G is the d-dimension...

In this work we propose a representation of the web as a directed hypergraph, instead of a graph, where links can connect not only pairs of pages, but also pairs of disjoint sets of pages. In our model, 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. Each hyperarc ...

We generalize the notion of an Euler tour in a graph in the following way. An Euler family in a hypergraph is a family of closed walks that jointly traverse each edge of the hypergraph exactly once. An Euler tour thus corresponds to an Euler family with a single component. We provide necessary and sufficient conditions for the existence of an Euler family in an arbitrary hypergraph, and in part...

K-way hypergraph partitioning has an ever-growing use in parallelization of scientific computing applications. We claim that hypergraph partitioning with multiple constraints and fixed vertices should be implemented using direct K-way refinement, instead of the widely adopted recursive bisection paradigm. Our arguments are based on the fact that recursive-bisection-based partitioning algorithms...

Abstract With the development of deep learning on high-order correlations, hypergraph neural networks have received much attention in recent years. Generally, can be divided into two categories, including spectral-based methods and spatial-based methods. For methods, convolution operation is formulated spectral domain graph, we introduce typical (HGNN), with (Hyper-Atten), hyperbolic network (H...

Abstract In this chapter, we introduce the mathematical foundations of hypergraph and present notations that are used to facilitate deep understanding analysis structure. A is composed a set vertices hyperedges, it generalization graph, where weighted quantifies relative importance hyperedges or vertices. Hypergraph can also be divided into two main categories, i.e., undirected representation d...