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.Comment: 13 pages, 5 figures. accepted by Theoretial Computer Scienc