On discrete gradient vector fields and Laplacians of simplicial complexes

TL;DR

This paper explores the relationship between discrete gradient vector fields and Laplacians of simplicial complexes, revealing their generating functions and characterizing their correspondence with rooted forests in higher dimensions.

Contribution

It proves that Laplacian characteristic polynomials generate discrete gradient vector fields and characterizes their correspondence with rooted forests in higher dimensions.

Findings

Laplacian characteristic polynomials are generating functions for discrete gradient vector fields.

Full characterization of the correspondence between rooted forests and discrete gradient vector fields.

Applicable to graphs and triangulations of orientable manifolds.

Abstract

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.