Matrix-tree theorems and discrete path integration

We calculate characteristic polynomials of operators explicitly presented as polynomials of rank 1 operators. Corollaries of the main result (Theorem 2.3) include a generalization of the Forman's formula for the determinant of the graph Laplacian [6,8], the celebrated Matrix-tree theorem by G. Kirchhoff [9], and some its extensions and analogs, both known (e.g. the Matrix-hypertree theorem by G. Masbaum and A. Vaintrob [10]) and new.