Graph theory formulation

Many researchers tried to explain the secret of the Clar s aromatic sextet theory, or hypothesis from quantum-chemical points of view. However, those trials have been failing until the graph and combinatorial theories came to be applied to this challenging problem [9,10]. In the following discussion it will be shown how various techniques and concepts of the graph theory are useful for realizing and formulating not only the fantastic theory of Clar but also the mathematical beauty of the structural formula of aromatic hydrocarbons. 

To repeat the route of chemistry in the kinetic aspect , that was the formulation of the problem. To our mind, however, in the 1930s "the rational classification principle , whose appearance was predicted by Semenov, could not be realized. The possibility of solving this problem appeared only in recent times in terms of the concepts of the graph theory and the qualitative theory of differential equations. The analysis of the effect of the mechanism structure on the kinetic regularities of catalytic reactions is one of the connecting subjects in the present study. 

There is no doubt that studies for the establishment of new classes of mechanisms possessing an unique and stable steady state are essential and promising. On the other hand, it is of interest to construct a criterion for uniqueness and multiplicity that would permit us to analyze any reaction mechanism. An important contribution here has been made by Ivanova [5]. Using the Clark approach [59], she has formulated sufficiently general conditions for the uniqueness of steady states in a balance polyhedron in terms of the graph theory. In accordance with ref. 5 we will present a brief summary of these results. As before, we proceed from the validity of the law of mass action and its analog, the law of acting surfaces. Let us also assume that a linear law of conservation is unique (the law of conservation of the amount of catalyst). 

Apart from enzyme kinetics, this new trend had also appeared in the kinetics of heterogeneous catalysis. In the 1950s, Horiuti formulated a theory of steady-state reactions [11, 12], many of the concepts of which correspond to the graph theory. Independent intermediates, a reaction route, an independent reaction route, all these concepts were introduced by Horiuti. 

The basic results in the analysis of non-linear mechanisms using graphs, were obtained by Clark [29], who developed a detailed formalism, and Ivanova [30, 31]. On the basis of Clark s approach, Ivanova formulated sufficiently general conditions for the uniqueness of steady states in terms of the graph theory. She suggested an algorithm that can be used to obtain (see Chap. 3, Sect. 5.4)... 

Now compare the conclusions deduced to what was said in Chapter 2 and you will see that what has just been presented is nothing else but the pairing theorem formulated in terms of the graph theory. 

Klein, D.J. (2003a) Graph theoretically formulated electronic-structure theory. Internet Electron. J. Mol. Des., 2, 814—834. 

In the formulation of Theorem 4.4.1.3 we have used the word unfortunately. This Is because the existence of isospectrai graphs indicates that a substantial part of the information about the structure of a gr h G has been lost in Ch(6). The theory of the characteristic polynomial is thus less rich than the theory of graphs and many problems in graph theory are simply intractable by means of graph spectral theory. As a consequence, the power of polynomial techniques in graph theory is much reduced. Similarly, several difficulties occur in the applications of Ch in chemistry. 

The problem to find the shortest path through a set of points can be formulated as the "travelling salesman problem . Suppose a salesman has to visit customers in several towns and he looks for the shortest way through all towns. An exact description of the problem and the solution requires some knowledge in graph theory C189, 240, 386, 4083. 

D.J. Klein, Graph theoretically formulated electronic-structure theory, Internet Electronic J. Mol. Des. 2 (2003) 814—834. http //www.hiochempress.com D.J. Klein and N. Trinajstic, Foundations of conjugated-circuit models. Pure Appl. Chem. 61 (1989) 2107-2115. 

The graph-theory approach is also able to formulate a simple theory for describing chemical reactions [10]. Let us consider two graphs Gi = (Vi, Ei, Li) and G2= (V2 B2 >L2) from the family 3pq We introduce a non-symmetric relation between these graphs called the chemical transformation... 

While most unit reactions are formulated graphically as monocycles of changing atoms in the reaction center, other contours for reaction centers have been recognized by graph theory and systematized by computer. Several of these approaches have led to the invention of new reactions and their... 

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]. 

Investigations with the graphs of non-linear mechanisms had been stimulated by an actual problem of chemical kinetics to examine a complex dynamic behaviour. This problem was formulated as follows for what mechanisms or, for a given mechanism, in what region of the parameters can a multiplicity of steady-states and self-oscillations of the reaction rates be observed Neither of the above formalisms (of both enzyme kinetics and the steady-state reaction theory) could answer this question. Hence it was necessary to construct a mainly new formalism using bipartite graphs. It was this formalism that was elaborated in the 1970s. 

The topic of the present section is a formulation of an alternative graph-theoretical model of synthons [16,24]. Such an approach makes it possible to build-up all concepts and notions of synthon theory [18-22] in a very transparent and easily understandable level. 

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