Bipartite graph

Topological matrix of a bipartite graph. Bipartite graphs, just as the corresponding alternant systems, possess a number of remarkable properties. In particular, their vertices can always be enumerated so that the topological matrix is simplified and reduced to the block form... 

The two cations, or two anions, that form the bond can be considered as a single pseudo-atom. This makes the bond graph bipartite. This solution works well for cations like Hg2 which are traditionally described in this way. In the mercurous cation, the Hg-Hg bond is formed by an electron pair and has a valence of 1.0 vu. If the Hg2 cation is in an asymmetric environment, the valence of the external bonds formed by the individual Hg atoms may not be the same, in which case the two mercury atoms may contribute different numbers of electrons to the Hg-Hg bond. Although this violates the equal contribution mle (3), the valence of the bond is correctly given by the average of the contributions of the two mercury atoms. Hg-Hg bonds are known in a number of mercury complexes, and not all of these are electron pair bonds, but as expected, the length of the bond is found to correlate with its valence (the number of electron pairs that form it) in the same way as any other bond [26]. 

Hitherto it has been assumed that the bond graph is bipartite, i.e. bonds only occur between a cation and an anion with no cation-cation or anion-anion bonds present. While the majority of inorganic compounds have bipartite bond graphs, there are a few, such as mercurous and peroxy compounds, that contain homoionic bonds. It is easy to see that there can be no electric flux linking two cations or two anions, so the ionic model predicts that no bond will exist between them. 

However, if the atoms are not related by symmetry, the normal rules break down. The homoionic N-N bond in the hydrazinium ion is an electron pair bond, but one in which N1 contributes 1.25 and N2 0.75 electrons. How can we apply the bond valence model in such cases where no solution to the network equations is possible One approach is to isolate the non-bipartite portion of the graph into a complex pseudo-atom. Thus in the hydrazinium ion the homoionic bond and its two terminating N atoms are treated as a single pseudo-anion which forms six bonds with a valence sum equal to the formal charge of —4. 

Since all the bonds in an inorganic compound with a bipartite graph start at a cation and end at an anion, they must obey the Coordination number rule. 

Conceptually, the representation of alternative process flowsheet(s) is based on elementary graph theory ideas. By representing each unit of the superstructure as a node, each input and output as a node, the interconnections among the process units as two-way arcs, the interconnections between the inputs and the process units as one-way arcs, the interconnections between the process units and the outputs as one-way arcs, and the interconnections between the inputs and the outputs as one-way arcs, then we have a bipartite planar graph that represents all options of the superstructure. 

Since a benzenoid system H is a bipartite graph, the existence of Kekule structures of H is equivalent to the existence of 1-factors (perfect matchings) of a bipartite graph. In 1935, P. Hall found the following necessary and sufficient conditions. 

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

Fig. 2. Examples of simple bipartite graphs, (a) Acyclic graph for the reaction Aj - A2 A (b) cyclic graph for the reaction A A2 (c) graph for the irreversible...

Fig. 3. Bipartite graphs for the mechanism of CO oxidation on Pt. (a) Eley-Rideal (impact) mechanism (b) Langmuir-Hinshelwood (adsorption) mechanism.

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). 

We will consider simple examples, i.e. the Eley-Rideal and Langmuir-Hinshelwood mechanisms for CO oxidation on Pt. Bipartite graphs corresponding to these mechanisms are represented in Fig. 3. In accordance with the general scheme, let us list segments, paths and cycles of these graphs. 

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. 

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