Witten Complex of Transitive Digraph and Its Convergence

Chong Wang; Xin Lai; Rongge Yu; Yaxuan Zheng; Baowei Liu

doi:doi:10.11648/j.mcs.20230802.12

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Witten Complex of Transitive Digraph and Its Convergence

Chong Wang, Xin Lai, Rongge Yu, Yaxuan Zheng, Baowei Liu

Published in Mathematics and Computer Science (Volume 8, Issue 2)

Received: 21 February 2023 Accepted: 24 March 2023 Published: 27 March 2023

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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), 2024. Published by Science Publishing Group

Keywords

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

References

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[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.

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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

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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

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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

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@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}
}

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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  -

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Author Information

Chong Wang

Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China
Xin Lai

School of Mathematics, Renmin University of China, Beijing, China
Rongge Yu

Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China
Yaxuan Zheng

Department of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, China
Baowei Liu

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

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Plain Text BibTeX RIS

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

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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

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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