Editing Hypergraph (section)

==Incidence matrix==
Let <math>V = \{v_1, v_2, ~\ldots, ~ v_n\}</math> and <math>E = \{e_1, e_2, ~ \ldots ~ e_m\}</math>. Every hypergraph has an <math>n \times m</math> [[incidence matrix]].

For an undirected hypergraph, <math>I = (b_{ij})</math> where
:<math>b_{ij} = \left\{ \begin{matrix} 1 & \mathrm{if} ~ v_i \in e_j \\ 0 & \mathrm{otherwise}. \end{matrix} \right.</math>
The [[transpose]] <math>I^t</math> of the [[incidence (geometry)|incidence]] matrix defines a hypergraph <math>H^* = (V^*,\ E^*)</math> called the '''dual''' of <math>H</math>, where <math>V^*</math> is an ''m''-element set and <math>E^*</math> is an ''n''-element set of subsets of <math>V^*</math>. For <math>v^*_j \in V^*</math> and <math>e^*_i \in E^*, ~ v^*_j \in e^*_i</math> [[if and only if]] <math>b_{ij} = 1</math>.

For a directed hypergraph, the heads and tails of each hyperedge <math>e_j</math> are denoted by <math>H(e_j)</math> and <math>T(e_j)</math> respectively.<ref name=gallo>{{cite journal |first1=G. |last1=Gallo |first2=G. |last2=Longo |first3=S. |last3=Pallottino |first4=S. |last4=Nguyen |title=Directed hypergraphs and applications |journal=Discrete Applied Mathematics |volume=42 |issue=2–3 |pages=177–201 |year=1993 |doi=10.1016/0166-218X(93)90045-P |doi-access=free }}</ref> <math>I = (b_{ij})</math> where
:<math>b_{ij} = \left\{ \begin{matrix} -1 & \mathrm{if} ~ v_i \in T(e_j) \\ 1 & \mathrm{if} ~ v_i \in H(e_j) \\ 0 & \mathrm{otherwise}. \end{matrix} \right.</math>

===Incidence graph===
A hypergraph ''H'' may be represented by a [[bipartite graph]] ''BG'' as follows: the sets ''X'' and ''E'' are the parts of ''BG'', and (''x<sub>1</sub>'',  ''e<sub>1</sub>'') are connected with an edge if and only if vertex ''x<sub>1</sub>'' is contained in edge ''e<sub>1</sub>'' in ''H''.

Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. This bipartite graph is also called [[incidence graph]].