The graph Laplacian and Morse inequalities

Abstract

The objective of this note is to provide an interpretation of the discrete version of Morse inequalities, following Witten's approach via supersymmetric quantum mechanics [8], adapted to finite graphs, as a particular instance of Morse-Witten theory for cell complexes [4]. We describe the general framework of graph quantum mechanics and we produce discrete versions of the Hodge theorems and energy cut-offs within this formulation.

Laplace and Dirac Operators on Graphs

“…Spinors have natural linear transformations, which can be seen as rotations in R 3 . More generally, it turns out that quaternions are useful to describe higher dimensional rotations, and therefore, spinors.…”

Graph de Rham Cohomology and the Automorphsim Group

“…Along this direction, the first author and Xu [3] provided a graph-theoretical version of Witten's approach to Morse inequalities [17], using supersymmetric quantum mechanics.…”