Topological Graphs, Structures

Another approach applies graph theory. The analogy between a structure diagram and a topological graph is the basis for the development of graph theoretical algorithms to process chemical structure information [33-35]. 

The only information that structural formulas provide is how a given atom is bonded (if at all) to others, or, in the words of Butlerov, how the chemical interaction is distributed . Moreover, when chemists began using the graph theory, they simplified the representation of a molecule as compared to the classical one. In distinction to a structural formula a topological graph, as a rule, has no... 

From the Lipscomb theory one can get the topological graphs for any borane of assigned composition and thus predict new structures. Certain limitations should be imposed, though, on the quantities appearing in the balance equations (31)-(34), for example,... 

Fig. 6. Uranyl selenate units observed in the structures of organic-inorganic nanocomposite compounds (a)LUS-l (b)LUS-2 andLUS-4 (c)LUS-3 andLUS-5. Below polyhedral diagrams black-and-white graphs describing topology of structural units are given.

AEI indices are atomic descriptors calculated from augmented matrices + P and + P derived from a topological graph representing the structure of valence electrons of atoms, where vertices are the valence electrons and edges indicate interactions between valence electrons [Li, Dai et al., 2005]. 

Ivanciuc, O. (2001c) Design of topological indices. Part 19. Computation of vertex and molecular graph structural descriptors with operators. Rev. Roum. Chim., 46, 243-253. 

Topological graphs are concerned only with which nodes (atoms) are joined by which edges (bonds), thus have difficulty in representing stereochemistry directly. A number of approaches have been used to incorporate information about stereochemical configuration into chemical structure representations. ... 

When a molecule is presented as more than one topological graph, the individual representations are termed mesomers or, in the context of structure registration, mesomeric duplicates. Mesomers differ only in bond order and placement of formal charges and are used to represent bond delocalisation in a molecule. 

For some appUcations it suffices to use descriptors that describe the topology of structures only (see e.g. Section 7.4). However, there ene many molecular properties that depend on the 3D shape of the molecules (see Section 7.5). Descriptors that take this information into account are called geometrical descriptors or geometrical indices. To allow calculation of geometrical descriptors, the atoms of the molecular graph M M have to be given 3D coordinates (iR ) (see Subsection 2.63). Various methods are available for this task [271,272], we use an empirical force field method similar to [5]. 

Many of the descriptors which can be calculated from the 2D structure rely upon the molecular graph representation because of the need for rapid calculations. Kier and Hall have developed a large number of topological indices, each of which characterises the molecular structure as a single number [Hall and Kier 1991]. Every non-hydrogen atom ir the molecule is characterised by two delta values, the simple delta Si and the valence delta SJ ... 

Most of the 2D QSAR methods are based on graph theoretic indices, which have been extensively studied by Randic [29] and Kier and Hall [30,31]. Although these structural indices represent different aspects of molecular structures, their physicochemical meaning is unclear. Successful applications of these topological indices combined with multiple linear regression (MLR) analysis are summarized in Ref. 31. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least square (PLS) [32] analysis has been employed [33]. 

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