| Peer-Reviewed

Witten Complex of Transitive Digraph and Its Convergence

Received: 21 February 2023     Accepted: 24 March 2023     Published: 27 March 2023
Views:       Downloads:
Abstract

Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.

Published in Mathematics and Computer Science (Volume 8, Issue 2)
DOI 10.11648/j.mcs.20230802.12
Page(s) 46-50
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Transitive Digraph, Discrete Morse Function, Witten-Morse Function, Path Homology, Witten Complex

References
[1] R. Ayala, L. M. Fernández and J. A. Vilches, Discrete Morse inequalities on infinite graphs. Electron. J. Combin. 16 (1) (2009), R38.
[2] R. Ayala, L. M. Fernández and J. A. Vilches, Morse inequalities on certain infinite 2-complexes. Glasg. Math. J. 49 (2) (2007), 155-165.
[3] R. Ayala, L. M. Fernández, D. Fernández-Ternero and J. A. Vilches. Discrete Morse theory on graphs. Topol. Appl. 156 (2009), 3091-3100.
[4] R. Ayala, L. M. Fernández, A. Quintero and J. A. Vilches, A note on the pure Morse complex of a graph. Topol. Appl. 155 (2008), 2084-2089.
[5] R. Forman, Morse theory for cell complexes. Adv. Math. 134 (1998), 90-145.
[6] R. Forman, A user’s guide to discrete Morse theory. Sém. Lothar. Combin 48 (2002), 35pp.
[7] R. Forman, Discrete Morse theory and the cohomology ring. Trans. Amer. Math. Soc. 354 (12) (2002), 5063- 5085.
[8] R. Forman, Witten-Morse theory for cell complexes. Topology 37 (5) (1998), 945-979.
[9] A. Grigor’yan, Y. Lin, Y. Muranov and S. T. Yau, Homologies of path complexes and digraphs. Preprint arXiv: 1207. 2834v4 (2013).
[10] A. Grigor’yan, Y. Lin, Y. Muranov and S. T. Yau, Homotopy theory for digraphs. Pure Appl. Math. Q. 10 (4) (2014), 619-674.
[11] A. Grigor’yan, Y. Lin, Y. Muranov and S. T. Yau, Cohomology of digraphs and (undirected) graphs. Asian J. Math. 19 (5) (2015), 887-932.
[12] A. Grigor’yan, Y. Lin, Y. Muranov and S. T. Yau, Path complexes and their homologies, preprint (2015), https://www.math.uni-bielefeld.de/grigor/dnote.pdf. to appear in Int. J. Math.
[13] A. Grigor’yan, Y. Muranov and S. T. Yau, Homologies of digraphs and Künneth formulas. Commun. Anal. Geom. 25 (5) (2017), 969-1018.
[14] A. Grigor’yan, Y. Muranov, V. Vershinin and S. T. Yau, path homology theory of multigraphs and quivers. Forum Math. 30 (5) (2018), 1319-1337.
[15] A. Grigor’yan, R. Jimenez, Y. Muranov and S. T. Yau, Homology of path complexes and hypergraphs. Topol. Appl. 267 (2019), 106877, in press.
[16] A. Grigor’yan, Y. Lin and S. T. Yau, Torsion of digraphs and path complexes. arXiv: 2012.07302v1.
[17] Yong Lin, Chong Wang, Shing-Tung Yau, Discrete Morse Theory on digraphs. Pure and Applied Mathematics Quarterly. 17 (5) (2021), 1711-1737.
[18] Jeremy van der Heijden, Morse Theory and Supersymmetry. Universiteit van Amsterdam. July 1, 2016.
[19] C. Wang, S. Ren, A Discrete Morse Theory for Digraphs. arXiv: 2007.13425.
[20] C. Wang, Discrete Morse functions on digraphs and their equivalence. Acta Scientiarum Naturalium Universitatis Nankaiensis. Accepted.
Cite This Article
  • APA Style

    Chong Wang, Xin Lai, Rongge Yu, Yaxuan Zheng, Baowei Liu. (2023). Witten Complex of Transitive Digraph and Its Convergence. Mathematics and Computer Science, 8(2), 46-50. https://doi.org/10.11648/j.mcs.20230802.12

    Copy | Download

    ACS Style

    Chong Wang; Xin Lai; Rongge Yu; Yaxuan Zheng; Baowei Liu. Witten Complex of Transitive Digraph and Its Convergence. Math. Comput. Sci. 2023, 8(2), 46-50. doi: 10.11648/j.mcs.20230802.12

    Copy | Download

    AMA Style

    Chong Wang, Xin Lai, Rongge Yu, Yaxuan Zheng, Baowei Liu. Witten Complex of Transitive Digraph and Its Convergence. Math Comput Sci. 2023;8(2):46-50. doi: 10.11648/j.mcs.20230802.12

    Copy | Download

  • @article{10.11648/j.mcs.20230802.12,
      author = {Chong Wang and Xin Lai and Rongge Yu and Yaxuan Zheng and Baowei Liu},
      title = {Witten Complex of Transitive Digraph and Its Convergence},
      journal = {Mathematics and Computer Science},
      volume = {8},
      number = {2},
      pages = {46-50},
      doi = {10.11648/j.mcs.20230802.12},
      url = {https://doi.org/10.11648/j.mcs.20230802.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230802.12},
      abstract = {Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.},
     year = {2023}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Witten Complex of Transitive Digraph and Its Convergence
    AU  - Chong Wang
    AU  - Xin Lai
    AU  - Rongge Yu
    AU  - Yaxuan Zheng
    AU  - Baowei Liu
    Y1  - 2023/03/27
    PY  - 2023
    N1  - https://doi.org/10.11648/j.mcs.20230802.12
    DO  - 10.11648/j.mcs.20230802.12
    T2  - Mathematics and Computer Science
    JF  - Mathematics and Computer Science
    JO  - Mathematics and Computer Science
    SP  - 46
    EP  - 50
    PB  - Science Publishing Group
    SN  - 2575-6028
    UR  - https://doi.org/10.11648/j.mcs.20230802.12
    AB  - Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.
    VL  - 8
    IS  - 2
    ER  - 

    Copy | Download

Author Information
  • Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China

  • School of Mathematics, Renmin University of China, Beijing, China

  • Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China

  • Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China

  • Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China

  • Sections