A hypergraphMathworldPlanetmath is an ordered pair (V,) where V is a set of vertices and is a set of edges such that 𝒫(V). In other words, an edge is nothing more than a set of vertices. A hypergraph is the same thing as a simple incidence structure, but with terminology that focuses on the relationship with graphs.

Sometimes it is desirable to restrict this definition more. The empty hypergraph is not very interesting, so we usually accept that V. Singleton edges are allowed in general, but not the empty one. Most applications consider only finite hypergraphs, but occasionally it is also useful to allow V to be infiniteMathworldPlanetmath.

Many of the definitions of graphs carry verbatim to hypergraphs.

is said to be k-uniform if every edge e has cardinality k, and is uniform if it is k-uniform for some k. An ordinary graph is merely a 2-uniform hypergraph.

The degree of a vertex v is the number of edges in that contain this vertex, often denoted d(v). is k-regularPlanetmathPlanetmathPlanetmathPlanetmath if every vertex has degree k, and is said to be regular if it is k-regular for some k.

Let V={v1,v2,,vn} and ={e1,e2,,em}. Associated to any hypergraph is the n×m incidence matrix A=(aij) where

aij={1 if viej0 otherwise 

For example, let =(V,), where V={a,b,c} and ={{a},{a,b},{a,c},{a,b,c}}. Defining vi and ej in the obvious manner (as they are listed in the sets), we have


Notice that the sum of the entries in any column is the cardinality of the corresponding edge. Similarly, the sum of the entries in a particular row is the degree of the corresponding vertex.

The transpose At of the incidence matrix also defines a hypergraph *, the dual of , in an obvious manner. To be explicit, let *=(V*,*) where V* is an m-element set and * is an n-element set of subsets of V*. For vj*V* and ei**,vj*ei* if and only if aij=1.

Continuing from the example above, the dual * of consists of V*={x,y,z,t} and *={{x,y,z,t},{y,t},{z,t}}, where vj* and ei* are defined in the obvious manner.

By the remark immediately after the definition of the incidence matrix of a hypergraph, it is easy to see that the dual of a uniform hypergraph is regular and vice-versa. It is not rare to see fruitful results emerge by considering the dual of a hypergraph.

Note A finite hypergraph is also related to a metacategory, and can be shown to be a special case of a supercategoryPlanetmathPlanetmath, and can be thus defined as a mathematical interpretationMathworldPlanetmathPlanetmath of ETAS axioms.

Title hypergraph
Canonical name Hypergraph
Date of creation 2013-03-22 13:05:29
Last modified on 2013-03-22 13:05:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 24
Author CWoo (3771)
Entry type Definition
Classification msc 05C65
Synonym metacategory
Synonym multi-graph
Synonym colored graph
Related topic SteinerSystem
Related topic IncidenceStructures
Related topic ToposAxioms
Related topic TopicEntryOnTheAlgebraicFoundationsOfMathematics
Related topic GraphTheory
Related topic ETAS
Related topic AxiomsOfMetacategoriesAndSupercategories
Defines incidence matrix
Defines uniform hypergraph
Defines regular hypergraph