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While graph edges are 2-element subsets of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a ''k-uniform hypergraph'' is a hypergraph such that all its hyperedges have size ''k''. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting ''k'' nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. A hypergraph is also called a ''set system'' or a ''[[family of sets]]'' drawn from the [[universal set]].  

While graph edges are 2-element subsets of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a ''k-uniform hypergraph'' is a hypergraph such that all its hyperedges have size ''k''. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting ''k'' nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. A hypergraph is also called a ''set system'' or a ''[[family of sets]]'' drawn from the [[universal set]].  

普通图的边是节点的二元子集，超边则是节点的任意集合，所以可以包含任意数量的节点。但我们总先需要研究具有相同基数超边的超图，即一个'''k-均匀超图 k-uniform hypergraph'''（所有超边的大小都为 k）。因此一个2-均匀超图就是普通图，一个3-均匀超图就是无序三元组 unordered triples 的集合，依此类推。超图也被称为从泛集 universal set 中抽取的一个集系统 set system 或集族 family of sets。

普通图的边是节点的二元子集，超边则是节点的任意集合，所以可以包含任意数量的节点。但我们总先需要研究具有相同基数超边的超图，即一个'''k-一致超图 k-uniform hypergraph'''（所有超边的大小都为 k，也译作k-均匀超图）。因此一个2-一致超图就是普通图，一个3-一致

 

超图就是无序三元组 unordered triples 的集合，依此类推。超图也被称为从泛集 universal set 中抽取的一个集系统 set system 或集族 family of sets。

Hypergraphs can be viewed as [[incidence structure]]s.  In particular, there is a bipartite "incidence graph" or "[[Levi graph]]" corresponding to every hypergraph, and conversely, most, but not all, [[bipartite graph]]s can be regarded as incidence graphs of hypergraphs.

Hypergraphs can be viewed as [[incidence structure]]s.  In particular, there is a bipartite "incidence graph" or "[[Levi graph]]" corresponding to every hypergraph, and conversely, most, but not all, [[bipartite graph]]s can be regarded as incidence graphs of hypergraphs.

The collection of hypergraphs is a [[Category (mathematics)|category]] with hypergraph [[homomorphism]]s as [[morphism]]s.

The collection of hypergraphs is a [[Category (mathematics)|category]] with hypergraph [[homomorphism]]s as [[morphism]]s.

特殊类型的超图包括：上文简单讨论过的k-均匀超图；散簇 clutter，也即没有一条边作是另一条边的子集；以及'''抽象单纯复形 abstract simplicial complexes'''，这里面包含每条边的所有子集。

特殊类型的超图包括：上文简单讨论过的k-一致超图；散簇 clutter，也即没有一条边作是另一条边的子集；以及'''抽象单纯复形 abstract simplicial complexes'''，这里面包含每条边的所有子集。

超图是一个以超图同态 homomorphism 为'''态射 morphism '''的范畴。

超图是一个以超图同态 homomorphism 为'''态射 morphism '''的范畴。

* 简单超图 Simple hypergraph ：不包含循环和重复边的超图。

* 简单超图 Simple hypergraph ：不包含循环和重复边的超图。

* <math>k </math>-均匀超图 <math>k </math>-uniform hypergraph：每条超边都正好包含 k 个顶点的超图。

* <math>k </math>-一致超图 <math>k </math>-uniform hypergraph：每条超边都正好包含 k 个顶点的超图。

* <math>d </math>-正则超图 <math>d </math>-regular hypergraph：每个顶点的度数都是 𝑑 的超图。

* <math>d </math>-正则超图 <math>d </math>-regular hypergraph：每个顶点的度数都是 𝑑 的超图。

The<math>r(H)</math> of a hypergraph <math>H</math> is the maximum cardinality of any of the edges in the hypergraph.  If all edges have the same cardinality ''k'', the hypergraph is said to be ''uniform'' or ''k-uniform'', or is called a ''k-hypergraph''.  A graph is just a 2-uniform hypergraph.

The<math>r(H)</math> of a hypergraph <math>H</math> is the maximum cardinality of any of the edges in the hypergraph.  If all edges have the same cardinality ''k'', the hypergraph is said to be ''uniform'' or ''k-uniform'', or is called a ''k-hypergraph''.  A graph is just a 2-uniform hypergraph.

超图<math>H</math>的<math>r(H)</math>表示该超图中任何一条边的最大'''基数'''。如果所有边具有相同的基数''k''，则称该超图为均匀的或k-均匀的，或称之为k-超图。普通图只是一个2-均匀的超图。

超图<math>H</math>的<math>r(H)</math>表示该超图中任何一条边的最大'''基数'''。如果所有边具有相同的基数''k''，则称该超图为一致的或k-一致的，或称之为k-超图。普通图只是一个2-一致的超图。

The degree ''d(v)'' of a vertex ''v'' is the number of edges that contain it. ''H'' is ''k-regular'' if every vertex has degree ''k''.

The degree ''d(v)'' of a vertex ''v'' is the number of edges that contain it. ''H'' is ''k-regular'' if every vertex has degree ''k''.

The dual of a uniform hypergraph is regular and vice versa.

The dual of a uniform hypergraph is regular and vice versa.

Two vertices ''x'' and ''y'' of ''H'' are called ''symmetric'' if there exists an automorphism such that <math>\phi(x)=y</math>.  Two edges <math>e_i</math> and <math>e_j</math> are said to be  ''symmetric'' if there exists an automorphism such that <math>\phi(e_i)=e_j</math>.

Two vertices ''x'' and ''y'' of ''H'' are called ''symmetric'' if there exists an automorphism such that <math>\phi(x)=y</math>.  Two edges <math>e_i</math> and <math>e_j</math> are said to be  ''symmetric'' if there exists an automorphism such that <math>\phi(e_i)=e_j</math>.

许多涉及图的定理和概念也适用于超图，典型的例子有[[拉姆西定理]]（Ramsey's theorem）和超图的线图。研究图的对称性的一些方法也被扩展到超图。

许多涉及图的定理和概念也适用于超图，典型的例子有[[拉姆西定理]]（Ramsey's theorem）和超图的线图。研究图的对称性的一些方法也被扩展到超图。

均匀超图上有[[Erdős-Ko-Rado theorem]]和[[Kruskal-Katona theorem]]两个著名定理。

一致超图上有[[Erdős-Ko-Rado theorem]]和[[Kruskal-Katona theorem]]两个著名定理。

==Hypergraph drawing==

==Hypergraph drawing==