Benzenoids, modeling graphs

The mathematical chemist will recognize a as one of the Kekule valence-bond structures of the hydrocarbon, while is the corres nd-ing molecular graph Model is called an inner dual or dualist [2], is a caterpillar tree [3], and is called a Clar graph [4]. The latter two models are apparently quite different from the original skeleton, however, as it will turn out later, the topological properties of this benzenoid system are best modeled by either d or e-... 

Another conclusion may also be drawn. This strong dependence of REC values (as well as formerly stated EN and HOMA values) of benzene rings in the benzenoid hydrocarbons on topology may be of a key importance for understanding of the successful treatments of benzenoid hydrocarbons by the so-called graph-topological models [27]. 

There is another neat scheme (based on the linear recursion) applicable for hand computation on polyhex benzenoids of up to a dozen or so hexagonal rings. This John-Sachs scheme is based on a one-to-one correspondence between Kekule structures and sets of mutually self-avoiding directed walks on the graph, and indeed this correspondence was (in a special context) utilized in a statistical mechanical context in modelling collections of partly disordered polymer chains. 

What keeps various graph theoretical and semi-empirical VB models alive is that the concepts involved in these models are expressed by a language that is commonly understood in chemistry, which includes such basic structural elements as atoms, bonds, rings, and various molecular fragments, such as the bay region or the pharmacophores, which are of interest in structure—bioactivity studies. In this article we will focus on aromaticity in hydrocarbons, and in particular the aromaticity of benzenoid hydrocarbons, and will see that there is some theoretical justification to expect the fully benzenoid 6n 7r-elec-tron systems of Clar to be the most aromatic polycyclic hydrocarbons. The reason that we have confined our attention to conjugated hydrocarbons is... 

Interpretation of statistical and other data on molecules. In this article in particular we will see how graph theoretical concepts, such as the conjugated circuits, the innate degree of freedom of Kekule valence structures, and the Clar structures, can all be combined to characterize the local and overall aromaticity of benzenoid hydrocarbons, fully benzenoid hydrocarbons, and fully aromatic compounds. In addition, we will see that these approaches that lead to the quantification of Clar s r-sextets model can be justified using chemical arguments. ... 

It may be of interest that isoconjugate benzenoid hydrocarbons necessarily have the same RE in Herndon s resonance theory VB model as in the conjugated circuits model, but generally they will have different RE in the HMO model and the TRE model, mentioned earlier. This clearly points to not only numerical but also conceptual differences between different graph theoretical models those based on the adjacency matrix (HMO) and that based on Kekule valence structures and conjugated circuits (VB). 

---