The Grassmann-Berezin calculus and theorems of the matrix-tree type

We prove two generalizations of the matrix-tree theorem. The first one, a result essentially due to Moon for which we provide a new proof, extends the "all minors" matrix-tree theorem to the "massive" case where no condition on row or column sums is imposed. The second generalization, which is new, extends the recently discovered Pfaffian-tree theorem of Masbaum and Vaintrob into a "hyper-Pfaffian-cactus" theorem. Our methods are noninductive, explicit and make critical use of the Grassmann-Berezin calculus that was developed for the needs of modern theoretical physics. (C) 2003 Elsevier Inc. All rights reserved.