Replacing Pfaffians and applications

We present some new Pfaffian identities related to the Plücker relations. As consequences we obtain a quadratic identity for the number of perfect matchings of plane graphs, which has a simpler form than the formula by Yan et al. [W.G. Yan, Y.-N. Yeh, F.J. Zhang, Graphical condensation of plane graphs: A combinatorial approach, Theoret. Comput. Sci. 349 (2005) 452–461], and we also obtain some new determinant identities.