PlanetPhysics/Hypergraph

A hypergraph or [[../Cod/|metagraph]] $\mathscr{H}$ is an ordered pair, or couple, $(V,ℰ)$ where $V$ is the class of vertices of the hypergraph and $ℰ$ is the class of edges such that $ℰ\subseteq 𝒫(V)$, where $𝒫(V)$ is the powerset of $V$ (the set of subsets of $V$) and is also considered to be a class.

A hypergraph is as an extension of the [[../PreciseIdea/|concepts]] of a [[../Cod/|graph]], colored graph and multi-graph. A finite hypergraph, with both $V$ and $ℰ$ being sets, is also related to a [[../Cod/|metacategory]]; therefore, it can also be considered as a special case of a [[../SuperCategory6/|supercategory]], and can be thus defined as a mathematical interpretation of [[../ETACAxioms/|ETAS axioms]].

A finite hypergraph can also be considered as an example of a simple incidence structure . Note also that the more general definition of a hypergraph given above avoids well known antimonies of set theory involving `sets' of sets in the general case.

Many specific graph definitions (but not all) can be extended to similar specific hypergraph, or multigraph, definitions. For example, let $V=\{v_1,v_2,\dots ,v_n\}$ and $ℰ=\{e_1,e_2,\dots ,e_m\}$. Associated to any finite hypergraph is the finite $n\times m$ incidence [[../Matrix/|matrix]] $A=(a_{ij})$ where

$a_{ij}=\{\begin{array}{cc}1& =if=v_i\in e_j\\ 0& =otherwise=\end{array}$

For example, let $\mathscr{H}=(V,ℰ)$, where $V=\{a,b,c\}$ and $ℰ=\{\{a\},\{a,b\},\{a,c\},\{a,b,c\}\}$. Defining $v_i$ and $e_j$ in the obvious manner (as they are listed in the sets), we have $A=\left(\begin{array}{cccc}1& 1& 1& 1\\ 0& 1& 0& 1\\ 0& 0& 1& 1\end{array}\right)$