Skip to main contentSkip to search
Episciences
Open Access Journals
Sign in(new window)
Discrete Mathematics & Theoretical Computer Science logo
Discrete Mathematics & Theoretical Computer Science
Discrete Mathematics & Theoretical Computer Science logo
Discrete Mathematics & Theoretical Computer Science
Sign in(new window)
Articles & Issues
All articlesAll accepted articlesAll volumesLast volumeSectionsSpecial issuesAuthors
About
The journalIndexing
Boards
Publish
For authorsEthical charter
Submit
Discrete Mathematics & Theoretical Computer Science logo
Contact
|
Credits
eISSN 1365-8050
|
RSS
|
Atom
Episciences
Documentation
|
Acknowledgements
|
Publishing policy
Accessibility: non-compliant
|
Legal mentions
|
Privacy statement
|
Terms of use
  1. Home > Articles & Issues >
  2. Articles >
  3. Critical Groups of S ...
Conference paper

Critical Groups of Simplicial Complexes

Art M. Duval (1), Caroline J. Klivans (2), Jeremy L. Martin ORCID (3)
(1) Department of Mathematical Sciences
(2) Department of Computer Science, University of Chicago
(3) Department of Mathematics
Download article
Open on HAL
Imported on
January 31, 2017
Published on
December 31, 2010
Last modified on
March 31, 2025
Volume
DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
Proceedings
DOI
10.46298/dmtcs.2909
License
https://about.hal.science/hal-authorisation-v1
Indicators
755
Views
1429
Downloads

Critical Groups of Simplicial Complexes

Art M. Duval (1), Caroline J. Klivans (2), Jeremy L. Martin ORCID (3)
(1) Department of Mathematical Sciences
(2) Department of Computer Science, University of Chicago
(3) Department of Mathematics
Abstract
English
We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups.
French
Nous généralisons la théorie des groupes critiques des graphes aux complexes simpliciaux. Plus précisément, pour un complexe simplicial, nous définissons une famille de groupes abéliens en termes d'opérateurs de Laplace combinatoires, qui généralise la construction du groupe critique d'un graphe. Nous montrons comment réaliser ces groupes critiques explicitement comme conoyaux des opérateurs de Laplace réduits combinatoires, et montrons qu'ils sont finis. Leurs ordres sont obtenus en comptant (avec des poids) des arbres simpliciaux couvrants. Nous décrivons comment les groupes critiques d'un complexe représentent le flux le long de ses faces, et esquissons une autre interprétation potentielle comme analogues des groupes de Chow.
Keywords
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • graph
  • simplicial complex
  • critical group
  • combinatorial Laplacian
  • chip-firing game
  • sandpile model
  • spanning trees
Cited by

Source: OpenCitations

  • Networks beyond pairwise interactions: Structure and dynamics

    Physics Reports

    Authors : Federico Battiston, Giulia Cencetti ORCID, Iacopo Iacopini ORCID, Vito Latora, Maxime Lucas ORCID, Alice Patania ORCID, Jean-Gabriel Young ORCID, Giovanni Petri ORCID

    Journal reference : Volume 874, 2020, pp. 1-92

    DOI : 10.1016/j.physrep.2020.05.004

  • The magnetic discrete Laplacian inferred from the Gauß–Bonnet operator and application

    Annals of Functional Analysis

    Authors : Nassim Athmouni, Hatem Baloudi, Mondher Damak ORCID, Marwa Ennaceur

    Journal reference : Volume 12, 2021, pp.

    DOI : 10.1007/s43034-021-00119-8

  • Abelian networks III: The critical group

    arXiv (Cornell University)

    Authors : Benjamin Bond, Lionel Levine ORCID

    Journal reference : Volume 43, 2015, pp. 635-663

    DOI : 10.1007/s10801-015-0648-4

  • A family of matrix-tree multijections

    Algebraic Combinatorics

    Authors : Alex McDonough ORCID

    Journal reference : Volume 4, 2021, pp. 795-822

    DOI : 10.5802/alco.181

  • Cuts and flows of cell complexes

    Journal of Algebraic Combinatorics

    Authors : Art M. Duval, Caroline J. Klivans, Jeremy L. Martin ORCID

    Journal reference : Volume 41, 2014, pp. 969-999

    DOI : 10.1007/s10801-014-0561-2

  • On the critical ideals of graphs

    arXiv (Cornell University)

    Authors : Hugo Corrales ORCID, Carlos E. Valencia ORCID

    Journal reference : Volume 439, 2013, pp. 3870-3892

    DOI : 10.1016/j.laa.2013.10.011

  • On Discrete Gradient Vector Fields and Laplacians of Simplicial Complexes

    Annals of Combinatorics

    Authors : Ivan Contreras ORCID, Andrew Tawfeek ORCID

    Journal reference : Volume 28, 2023, pp. 67-91

    DOI : 10.1007/s00026-023-00655-1

  • Chip Firing on General Invertible Matrices

    SIAM Journal on Discrete Mathematics

    Authors : Johnny Guzmán ORCID, Caroline Klivans

    Journal reference : Volume 30, 2016, pp. 1115-1127

    DOI : 10.1137/15m1034404

  • Simplicial and cellular trees

    ˜The œIMA volumes in mathematics and its applications

    Authors : Art M. Duval, Caroline J. Klivans, Jeremy L. Martin ORCID

    Journal reference : Volume , 2016, pp. 713-752

    DOI : 10.1007/978-3-319-24298-9_28

  • Links in surfaces and Laplacian modules

    arXiv (Cornell University)

    Authors : Daniel S. Silver ORCID, Susan G. Williams ORCID

    Journal reference : Volume 29, 2020, pp. 2050057

    DOI : 10.1142/s0218216520500571

  • Flow-firing processes

    arXiv (Cornell University)

    Authors : Pedro Felzenszwalb, Caroline Klivans

    Journal reference : Volume 177, 2020, pp. 105308

    DOI : 10.1016/j.jcta.2020.105308

  • Reviews

    American Mathematical Monthly

    Authors : Darren Glass

    Journal reference : Volume 127, 2020, pp. 189-192

    DOI : 10.1080/00029890.2020.1685835

  • Tropical complexes

    arXiv (Cornell University)

    Authors : Dustin Cartwright ORCID

    Journal reference : Volume 163, 2019, pp. 227-261

    DOI : 10.1007/s00229-019-01148-3

  • Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra

    arXiv (Cornell University)

    Authors : David Jekel, Avi Levy, Will Dana, Austin Stromme, Collin Litterell

    Journal reference : Volume 32, 2018, pp. 1040-1110

    DOI : 10.1137/16m1072607

  • Persistent Homology Apply in Digital Images

    Lecture notes on data engineering and communications technologies

    Authors : Sun Huadong ORCID, Zhang Yingjing, Zhao Pengfei

    Journal reference : Volume , 2021, pp. 392-400

    DOI : 10.1007/978-3-030-92632-8_37

  • Weighted spectra on a weighted geometric realization of 2-simplexes and 3-simplexes

    Discrete Mathematics Algorithms and Applications

    Authors : Azeddine Baalal ORCID, Khalid Hatim

    Journal reference : Volume 15, 2022, pp.

    DOI : 10.1142/s1793830922501300

  • Smith normal form in combinatorics

    arXiv (Cornell University)

    Authors : Richard P. Stanley ORCID

    Journal reference : Volume 144, 2016, pp. 476-495

    DOI : 10.1016/j.jcta.2016.06.013

  • Eigenvalues and critical groups of Adinkras

    Duo Research Archive (University of Oslo)

    Authors : Kevin Iga ORCID, Caroline Klivans, Jordan Kostiuk ORCID, Chi Ho Yuen

    Journal reference : Volume 143, 2022, pp. 102450

    DOI : 10.1016/j.aam.2022.102450
Preview
Loading PDF preview...