Vol’pert graph

Non-linear elementary steps can also be depicted by cyclic graphs when additional, secondary" edges are used. The nonlinear mechanisms can also be represented by so-called Vol pert graphs. ... 

Fig. 3.1 shows the Vol pert graph of the Lotka-Volterra reaction as an example. 

It may be illuminating that graphs used by the biochemist are either Vol pert graphs as in the case of the Lohmann reaction shown in Fig. 3.2 or are the duals of the Vol pert graph (and this is the more usual case) as... 

Hint. It is advantageous, although neither necessary nor sufficient if the Vol pert graph of the reaction has a tree structure. 

Let us consider a mechanism with an acyclic Vol pert graph, and let us suppose that the zero complex is not a reactant complex in any of the elementary reactions. Then... 

Clarke (1974a, b, 1980) gives a detailed analysis of the stability of the steady state using the structure of the Vol pert graph and several derived graphs to check the Routh-Hurwitz criterion. 

Show that the Vol pert graph of a weakly reversible mechanism is necessarily cyclic. 

Investigation of several examples shows that the deficiency of the models of these reactors is usually more than zero, and their Vol pert graph is usually cyclic. The authors mentioned above started another approach to avoid these problems. Their results belong to two different classes some of them come from the direct investigation of formally homogeneous models of CSTRs and others of them come from the application of the deficiency one theory. Before applying this theory it must (and can) be shown that chemically relevant assumptions define a class of models with deficiency one. 

A.I. Vol pert, E.A. Gel man and A.N. Ivanova, Some Problems of the Qualitative Theory of Differential Equations on Graphs, Preprint, OIKhF AN SSSR, Chernogolovka, 1975 (in... 

As the V-graph of a weakly reversible mechanism is cyclic (Exercise 4 below), the regular behaviour of a mechanism cannot be inferred from both zero deficiency theorem and by Vol pert s theorem at the same time at most one of them can be used. 

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... 

---