# The Graph Laplacian and Morse Inequalities

## TL;DR

This paper interprets discrete Morse inequalities using supersymmetric quantum mechanics on graphs, providing a framework that connects graph theory, Morse theory, and quantum mechanics with new discrete Hodge theorems.

## Contribution

It introduces a novel interpretation of Morse inequalities on graphs through supersymmetric quantum mechanics, extending Morse-Witten theory to finite graph structures.

## Key findings

- Discrete Morse inequalities are interpreted via graph quantum mechanics.
- Discrete versions of Hodge theorems are established.
- Energy cut-offs are formulated within the graph framework.

## 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, adapted to finite graphs, as a particular instance of Morse-Witten theory for cell complexes. 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.