Hypergraphs of molecular structure

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

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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 ... 

The results have shown that for considered series of molecular structures the indices sensitivity for the hypergraph model are higher than ones for the graph model. The data submitted below show the quantity of indices, the sensitivity Sq of which in the graph model is less, more, or equal to the sensitivity Sh of the indices in the hypergraph model ... 

So the comparative analysis of indices for both models of eight series of nonclassical molecular structures shows that the hypergraph model gives a higher accuracy of molecular structure description the higher accuracy of the model, the greater the diversity of the behavior of its invariants. 

E.V. Konstantinova, V.A. Skorobogatov, Graph and Hypergraph Models of Molecular Structure a Comparative Analysis of Indices, Zh.Strukt.Khim., 6 (1998) 958-966. 

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

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

Now we will give three examples to illustrate the difference between graphs and hypergraphs in the representation of the same molecular structures. 

Case 1. The shortest (u, i )-walk in graph G and hypergraph H consists only edges of degree two. In this case paths have the same length as G and H describe the same molecular structure F on the same set of vertices and as (u, u)-walks are the shortest. Thus... 

Figure 23. The example of obtaining the values of the Wiener index for hypergraph and graph representations of the molecular structure with three-electron ligands hydrogens are suppressed in these representations...

E. Konstantinova and V. Skorobogatov. Molecular hypergraphs The new Representation of nonclassical molecular structures with polycentric delocalized bonds./. Chem. Inf Comput. Sci., 35 472-478,1995. 

This example vividly illustrates the relationship between the structure of chemical compounds and the labels used in molecular hypergraphs. In particular, the labels of hyperedges E2 and Ei adequately reflect the essence of the metal-ligand bonds, in which ligands donate their electrons to form these bonds. 

In 1994 the concept of a molecular hypergraph, as a mathematical tool suitable for modeling the structure of molecules with polycentric bonds was introduced [46]. Main goal of this Lecture Notes is to demonstrate advantages of an application of hypergraph theory in chemical researches. 

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