Hypergraphs

A hypergraph is a generalization of a graph in which an edge can connect any number of vertices, rather than just two. This concept was introduced by Claude Berge in 1973. In a hypergraph, an edge is often referred to as a hyperedge, and it can be represented as a subset of the vertex set. Hypergraphs are used in various fields such as computer science, social network analysis, and operations research.

Hypergraphs can be represented using an incidence matrix, where rows correspond to vertices and columns correspond to hyperedges. An entry in the matrix is 1 if the vertex is incident to the hyperedge, and 0 otherwise. Another common representation is the bipartite graph, where one set of vertices represents the vertices of the hypergraph, and the other set represents the hyperedges.

Hypergraphs have several properties that are not present in simple graphs. For example, a hypergraph can have multiple edges connecting the same set of vertices, and it can also have loops, which are hyperedges that connect a single vertex to itself. Hypergraphs can also be directed, where the direction of the hyperedge is important.

Hypergraphs are used in various applications, such as in the study of social networks, where a hyperedge can represent a group of people who are all friends with each other. They are also used in operations research for modeling problems such as facility location and scheduling. In computer science, hypergraphs are used in the study of database systems and in the design of algorithms for graph problems.