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hypergraph (Definition)

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

Remark 0.1   A hypergraph is as an extension of the concepts of a graph, colored graph and multi-graph. A finite hypergraph, with both $V$ and $\mathcal{E}$ being sets, is also related to a metacategory; therefore, it can also be considered as a special case of a supercategory, and can be thus defined as a mathematical interpretation of ETAS axioms.
Remark 0.2   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.
Remark 0.3   Many specific graph definitions (but not all) can be extended to similar specific hypergraph, or multigraph, definitions. For example, let $V = \{v_1, v_2, ~\ldots, ~ v_n\}$ and $\mathcal{E} = \{e_1, e_2, ~ \ldots, ~ e_m\}$. Associated to any finite hypergraph is the finite $n \times m$ incidence matrix $A = (a_{ij})$ where

\begin{displaymath}a_{ij} = \begin{cases} 1 &\text{ if } ~ v_i \in e_j \ 0 &\text{ otherwise } \end{cases}\end{displaymath}
For example, let $\mathcal{H}=(V,\mathcal{E})$, where $V=\lbrace a,b,c\rbrace$ and $\mathcal{E}=\lbrace \lbrace a\rbrace, \lbrace a,b\rbrace, \lbrace a,c\rbrace, \lbrace a,b,c\rbrace\rbrace$. Defining $v_i$ and $e_j$ in the obvious manner (as they are listed in the sets), we have
$A = \begin{pmatrix} 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 \ 0 & 0 & 1 & 1 \end{pmatrix}$



"hypergraph" is owned by bci1.
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See Also: axioms of metacategories and supercategories, axiomatic theories of metacategories and supercategories

Other names:  simple incidence structure
Also defines:  finite hypergraph, hypergraph, simple incidence structure, incidence structure
Keywords:  finite hypergraph, hypergraph, simple incidence structure, incidence structur

Cross-references: matrix, ETAS axioms, supercategory, metacategory, graph, concepts, metagraph
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This is version 11 of hypergraph, born on 2010-05-09, modified 2010-05-10.
Object id is 865, canonical name is Hypergraph.
Accessed 1091 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 02.70.-cxx (Computational techniques )
 02.90.+p (Other topics in mathematical methods in physics )
Pending Errata and Addenda
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