Topological theory of graphs

Handling rate equations for complex mechanisms. While steady-state rate equations can be derived easily for the simple cases discussed in the preceding sections, enzymes are often considerably more complex and the derivation of the correct rate equations can be extremely tedious. The topological theory of graphs, widely used in analysis of electrical networks, has been applied to both steady-state and nonsteady-state enzyme kinetics 45-50 The method employs diagrams of the type shown in Eq. 9-50. Here... 

Topaquinone (TPQ) 816, 817s Topoisomerases 219,575, 638, 657,659 Topological theory of graphs 466 Topologies... 

We wish to conclude this paragraph with a quotation from the author s first paper [2] on benzenoid systems There is no simple recipe to decide by inspection of the molecular graph whether K = 0 or not. In other words, the necessary and sufficient conditions for the existence of Kekule structures seem to be rather complicated". Fifteen years later one may optimistically state that this hard problem of the topological theory of benzenoid hydrocarbons is completely settled. 

The graph-theoretical methods are extensively used in the theory of conjugated systems. It is only natural since in this area of chemistry the HMO method was used most frequently. In the next chapters we shall dwell on some interesting results derived in the topological theory of conjugated and aromatic systems. 

The central idea of this review is that the technique of topological reduction is a powerful one for the formal manipulation of cluster series and for generating useful approximate theoretical expressions for the thermodynamic properties and pair correlation functions of fluids. We have described just enough of the theory of graphs to have a meaningful discussion of topological reduction and have then discussed a number of types of theories in which topological reductions are applied. A detailed discussion of the approximate theories that result is outside the scope of this article, and the reader is referred to the literature cited in the text. 

S. El-Basil and D. J. Klein, Fibonacci numbers in the topological theory of benzenoid hydrocarbons and related graphs, J. Math. Chem. 3 (1989) 1-23. 

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]. 

Because of the fundamental role of graph theory for the understanding of topological descriptors, some terms from graph theory are defined below. Then a selection of topological descriptors are discussed. 

Mezey, P.G., Reaction Topology Manifold Theory of Potential Surfaces and Quantum Chemical Synthesis Design, in Chemical Applications of Topology and Graph Theory, Elsevier Sci. 

Schultz, H. P., Schultz, E. B., Schultz, T. P. Topological organic chemistry. 9. Graph theory and molecular topological indices of stereoisomeric organic compounds. J. Chem. Inf. Comput. Sci. 1995, 35, 864-870. 

More advanced mathematical aspects of the graph-theoretical models for aromaticity are given in other references [36, 48, 49]. Some alternative methods, beyond the scope of this chapter, for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [51-53] and the topological solutions of non-linear field theory related to the Skyrmions of nuclear physics [54]. 

King, B. (1993) Applications of Graph Theory and Topology in Inorganic Cluster and Coordination Chemistry (CRC Press Boca Raton, Ann Arbor). 

Trindle, C, Givan, R. In Chemical Applications of Graph Theory and Topology King, R. B, Ed. Elsevier New York, 1983. Trindle, C. Croatlca Chem. Acta 1984, 57, 1231,... 

To overcome this weakness, we are developing a quantitative structure-activity strategy that is conceptually applicable to all chemicals. To be applicable, at least three criteria are necessary. First, we must be able to calculate the descriptors or Independent variables directly from the chemical structure and, presumably, at a reasonable cost. Second, the ability to calculate the variables should be possible for any chemical. Finally, and most importantly, the variables must be related to a parameter of Interest so that the variables can be used to predict or classify the activity or behavior of the chemical (j ) One important area of research is the development of new variables or descriptors that quantitatively describe the structure of a chemical. The development of these indices has progressed into the mathematical areas of graph theory and topology and a large number of potentially valuable molecular descriptors have been described (7-9). Our objective is not concerned with the development of new descriptors, but alternatively to explore the potential applications of a group of descriptors known as molecular connectivity indices (10). 

The study of chemical reactions requires the definition of simple concepts associated with the properties ofthe system. Topological approaches of bonding, based on the analysis of the gradient field of well-defined local functions, evaluated from any quantum mechanical method are close to chemists intuition and experience and provide method-independent techniques [4-7]. In this work, we have used the concepts developed in the Bonding Evolution Theory [8] (BET, see Appendix B), applied to the Electron Localization Function (ELF, see Appendix A) [9]. This method has been applied successfully to proton transfer mechanism [10,11] as well as isomerization reaction [12]. The latter approach focuses on the evolution of chemical properties by assuming an isomorphism between chemical structures and the molecular graph defined in Appendix C. 

A topological analysis of the electron density in the framework of AIM theory, performed for the systems in Figure 6.2, has completely confirmed their formulation as dihydrogen-bonded complexes. In accord with the AIM criteria, the pc and V pc parameters at the bond critical points found in the H- - -H directions are typical of dihydrogen bonds 0.042 and 0.057 au for complex LiH HF and 0.046 and 0.048 au for complex NaH- - -HF, respectively. The presence of the bond critical points can be well illustrated by the molecular graph in Figure 6.3, obtained for the HCCH H-Li complex by Grabowski and co-workers [8]. 

The mathematical theory of topology is the basis of other approaches to understanding inorganic structure. As mentioned in Section 1.4 above, a topological analysis of the electron density in a crystal allows one to define both atoms and the paths that link them, and any description of structure that links pairs of atoms by bonds or bond paths gives rise to a network which can profitably be studied using graph theory. 

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