Enumerating $k$-matchings in cyclic chains using the transfer matrix technique

A cyclic chain is a plane graph whose all inner faces are cycles and its inner dual is isomorphic to a path. In this paper, the transfer matrix technique using the $k$-matching vector is developed to compute the number of $k$-matchings in an arbitrary cyclic chain. This extends similar methods developed earlier in two papers for benzenoid chains and for octagonal chains. The method is illustrated on the flourene molecule.