Transfer‐Matrix Method for Subgraph Enumeration: Applications to Polypyrene Fusenes

A frequently encountered problem in chemical applications is that of a weighted enumeration (or summation over) a class of extended subgraphs of a given system graph, which might represent a chemical structure. Some aspects of a powerful transfer‐matrix method are described for treating such graphtheoretic weighted enumeration problems. This method is seen to be particularly amenable for system graphs which are long in one direction and narrow in transverse directions. When the system graph is uniform (i.e., translationally symmetric) along one extended direction, asymptotic results can be readily extracted. A second point of emphasis here is that the weighted enumeration problems of the type studied here naturally arise in computing matrix elements over cluster expanded wave functions, though most applications so far framed in the literature differ from this. Size consistency and size‐extensivity aspects of this application are noted in terms of the transfer‐matrix approach. Polypyrene fusene strips of varying lengths are considered as applications of the transfer‐matrix methods for two weighted enumeration problems. Different graph‐theoretic problems are noted to arise for low‐order cluster expanded wave functions, such as in fact occur in both the Herndon‐Simpson and the Pauling‐Wheland resonance theories. For higher‐order wave function ansätze the graph‐theoretic problems would simply have more complicated weights and transfer matrices, which for the present examples are very small (i.e., 2 by 2 and 3 by 3).