Linear Recursive Methods - Kekule Structure Counting

2 Linear Recursive Methods - Kekule Structure Counting. - A rather 

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. 

The consideration of the sets of mutually self-avoiding walkers has a wider impact in revealing a fundamental invariant (the number of walkers) which turns out to be important in transfer-matrix solutions, as applied to polymer graphs extending even to the two-dimensional limit (as discussed in the next subsection). Also this invariant, which may also be described as an order, has physical implications.  

Overall the general recursion of eqn. (2) is applicable beyond the case of Kekule structures here elaborated for illustration. The related so-called conju-gated-circuit method turns out to have quite neat (related) linear re-cursions. Generally many sub-graph enumeration problems turn out to be of a linear recursive nature. 

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