Graphical condensation for enumerating perfect matchings

The method of graphical condensation for enumerating perfect matchings was found by Propp (Theoret. Comput. Sci. 303 (2003) 267), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004) 29). In this paper, we obtain some more general results on graphical condensation than Kuo's. Our method is also different from Kuo's. As applications of our results, we obtain a new proof of Stanley's multivariate version of the Aztec diamond theorem and we enumerate perfect matchings of a type of molecular graph. Finally, a combinatorial identity on the number of plane partitions is also given.