  Discrete Morse theory, a cell complex-analog to smooth Morse theory, has been
developed over the past few decades since its original formulation by Robin
Forman in 1998. In particular, discrete gradient vector fields on simplicial
complexes capture important features of discrete Morse functions. We prove that
the characteristic polynomials of the Laplacian matrices of a simplicial
complex are generating functions for discrete gradient vector fields of
discrete Morse functions when the complex is either a graph or a triangulation
of an orientable manifold. Furthermore, we provide a full characterization of
the correspondence between rooted forests in higher dimensions and discrete
gradient vector fields.
