Bipartite graph reaction mechanism

Fig. 1. Simple examples for bipartite graphs of reaction mechanisms. , Reaction nodes O, substance nodes.

In a certain sense, the simplest class of reaction mechanism is that whose bipartite graphs do not contain cycles, i.e. are acyclic. The dynamic behaviour of the corresponding reactions is always extremely simple [7]. An example for such a mechanism can be Ax - A2 - A3 - . . . - A [see Fig. 2(a)]. The contribution of acyclic mechanisms to the kinetics of catalytic reactions is not of importance. The mechanisms of catalytic reactions always contain cycles and these cycles are oriented, the directions of all the arrows being matched [the end of the ith arrow is the beginning of the... 

If all the elementary reactions are monomolecular, i.e. can be written as Ax —> Aj, it is more convenient to represent reaction mechanisms in a different way, namely nodes correspond to substances, edges are elementary reactions, and edge directions are the directions of reaction processes. As usual, this graph is simpler than the bipartite graph. For example, for the system of three isomers Al A2 and A3 we obtain... 

It can also be interpreted in terms of the bipartite graph for the reaction mechanism (see Sect. 1.3). 

On the basis of the structure for a bipartite graph of the reaction mechanism, it is possible to formulate a sufficient condition (174) for the uniqueness of a steady state. Applying it to concrete reactions, it is possible to establish the parametric areas for which either a unique steady state exists or there is a multiplicity of such states. 

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. 

Bipartite graphs for presenting complex mechanisms of chemical reactions have been proposed by Vol pert (1972) and Hudyaev and Vol pert (1985). These graphs contain nodes of two types type X nodes corresponding to components X, (/ = 1, 2,. ..,N) and type R nodes ascribed to... 

Examples of simple bipartite graphs for irreversible reactions (A) acyclic mechanism and (B) cyclic... 

In the analysis of bipartite graphs, the concept of cycles is crucial. The simplest class of reaction mechanisms is that with bipartite graphs that do not contain cycles (see Fig. 3.12A). These reaction mechanisms are called acyclic mechanisms and can be represented in general form as ... 

Some graph cycles may be nonoriented. Fig. 3.15 shows the bipartite graph for the two-step reaction mechanism... 

For the description of nonlinear mechanisms, several approaches have been proposed. Utilization of bipartite graphs for nonlinear kinetic mechanisms is one option, but could be very confusing even for simple reactions. 

An alternative way of reaction mechanism visualization is it to use bipartite graphs (BGs), which contain vertices of two types namely, those for the intermediates and steps. 

Fig. 4.21 Bipartite graph for the reaction mechanism in Fig. 4.20. (From O.N. Temkin, Kinetic models of multi-route reactions in homogeneous catalysis with metal complexes (a review), Kinet. Catal. 53 (2012) 313-343. Copyright 2012 Springer).

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