Vol. 300, No. 2, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 301: 1
Vol. 300: 1  2
Vol. 299: 1  2
Vol. 298: 1  2
Vol. 297: 1  2
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
 
Other MSP Journals
The graph Laplacian and Morse inequalities

Ivan Contreras and Boyan Xu

Vol. 300 (2019), No. 2, 331–345
Abstract

We provide an interpretation of the discrete version of Morse inequalities, following Witten’s approach via supersymmetric quantum mechanics (J. Differential Geom. 17:4 (1982), 661-692), adapted by Forman to finite graphs, as a particular instance of Morse–Witten theory for cell complexes (Topology 37:5 (1998), 945–979). 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.

Keywords
graph Laplacian, Morse–Witten complex, graph Hodge theory, discrete Morse functions
Mathematical Subject Classification 2010
Primary: 05C10, 81Q35
Secondary: 94C15
Milestones
Received: 15 May 2018
Revised: 16 October 2018
Accepted: 18 October 2018
Published: 30 July 2019
Authors
Ivan Contreras
Department of Mathematics and Statistics
Amherst College
Amherst, MA
United States
Boyan Xu
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States