Bipartite networks

As a matter of fact, the hole and particle occupancies are identical for any bipartite networks treated within Jt-approximation, up to FCl/PPP. This is a simple corollary of the generalized pairing theorem of McLachlan [94] stating that the jr-electron charge density matrix of the alternant hydrocarbons is of the form... 

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. 

Balandin (1970)16 appears to have used bipartite graphs to study reactions in the 1930s (though his work was not published until much later). Clarke (1980)17 used a type of bipartite graph which he termed a current diagram to study the stability of reaction networks. He found that for a reaction process to be... 

Fig. 6-16 A chemical-genetic network representing a graph C = (V, E) (data from [82]). Each node (V circles) represents a biologically active small molecule or a phenotypic assay and each edge (E line) represents an observed biological activity. Shown here is an undirected, unweighted, bipartite graph with a total of 426 nodes (V)...

A bond network with this property is said to have a bipartite graph, and a corollary to this definition is that the bond network of a valence compound crmtains only even-membered rings since an odd-membered ring necessarily contains a homoionic bond. 

This derivation of the ionic model provides not only a natural definition of a bond, but it also defines the scope of the model. The assumptions on which the derivation is based show that far from being confined to compotmds whose bonds have ionic character, the ionic model can be used for any valence compound. It can be used to describe not just NaCl but also SFs, CO2, CH4, CH3COOH, and O2, all of which have networks with bipartite graphs. 

The network equations (14a) and (14b) can only be used if the graph of the bond network is bipartite, that is, if every bond has a cation at one end and an anion at the other. In inorganic compounds, and particularly in organic compounds, this condition is not always satisfied. Although this restricts the application of the bond valence theory, the core-and-valence-shell picture of the atom is still valid, as is the description of the chemical bonds this picture gives. 

Bipartite graph The graph of a bond network in which the atoms are of two kinds (e.g., cation or anion) with no bonds occurring between atoms of the same kind Bond The chemical link between two neighboring atoms. In the ionic model, two atoms are bonded if and only if they are linked by electrostatic flux Bond network A topological description of the way in which atoms in a system are linked by bonds... 

Valence compound A compound having a bond network with a bipartite graph. 

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