Graphical condensation of plane graphs: a combinatorial approach

The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003), 267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57) and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this paper, by a purely combinatorial method some explicit identities on graphical vertex-condensation for enumerating perfect matchings of plane graphs (which do not need to be bipartite) are obtained. As applications of our results, some results on graphical edge-condensation for enumerating perfect matchings are proved, and we count the sum of weights of perfect matchings of weighted Aztec diamond.