Hypergraphs

Banchoff, T. (1990) From Flatland to Hypergraphics. Interdisciplinary Science Reviews. 15 364-72. 

John P, Rempel J (1985) Graphs, hypergraphs and applications, Proc. Conf. Graph Theory, Eyba, Oct. 1984, (Teubner-Texte zur Mathematik, Band 73) Teubner, Leipzig, p 72... 

All the defects of the structure representation that was mentioned above are eliminated if the hypergraphs are used to represent the structures with polycentric bonds [46, 47]. 

Before defining a molecular hypergraph, we give several notions of hypergraph theory which are necessarily for further discussion [8, 73]. 

An ordinary graph is a special case of a hypergraph with degrees of all the edges equal to two. 

The incidence matrix of hypergraph H is also shown in Figure 8. 

A hypergraph H = (V,E) is labeled if there exists labeling functions defined on the set of vertices V and on the set of edges E, which establish a correspondence between each vertex and each edge and some label. Let A V) = a(ui),...,a(up) and A E) = a Ei),..., a Eq) be the sets of possible labels called vertex and edge alphabets. The sets A V) and A E) may be represented by symbolic, numerical and structural labels. 

Let us give example of using permutations as hypergraph labels. Figure 10 shows a hypergraph whose set A E) of edge labels is represented by permutations 213 3 5 4... 

Figure 11. Molecular unlabeled (a-c) and labeled (d-f) hypergraphs representing allylic complex with different degrees of details...

The first way. Molecular structures are represented by the labeled hypergraphs whose vertex and edge labels are elements of the symbolic alphabets. 

Figure 13. Hypergraphs with the symbolic vertex and edge labels representing allylic and cyclopropenyl complexes given in Figure 12...

The vertex and edges alphabets of hypergraph H2 representing cyclopropenyl complex look like A V H2)) = CH, M and A E H2)) = [El, E2,E3,E4. The vertex labels are presented by symbols of functional chemical groups. It is easy to see that the vertex alphabets are the different for these hypergraphs. 

This way of hypergraph representation of molecular structures disregards the structure of ligands in the complexes. Yet it gives unambiguos descriptions, because the incidence matrices of the hypergraphs differ ... 

This example demonstrates the possibility of hypergraph modeling for molecular structure representation with the different degrees of accuracy. 

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