A long exact sequence of path homology of digraphs
-
Jian Liu
Abstract
In this paper, we develop a long exact sequence for the path homology of digraphs, providing a useful tool for computing the path homology of digraphs. One application of this result is the proof of a conjecture proposed by S. Chowdhury, which was initially observed through extensive computational experiments. Another interesting application demonstrates that the path homology of n-dimensional grid-like digraphs is concentrated in dimension
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12401080
Funding statement: This work was supported in part by the Natural Science Foundation of China (Grant No. 12401080) and the Scientific Research Foundation of Chongqing University of Technology.
References
[1] D. Chen, J. Liu, J. Wu, G.-W. Wei, F. Pan and S.-T. Yau, Path topology in molecular and materials sciences, J. Phys. Chem. Lett. 14 (2023), no. 4, 954–964. 10.1021/acs.jpclett.2c03706Suche in Google Scholar PubMed PubMed Central
[2] S. Chowdhury, S. Huntsman and M. Yutin, Path homologies of motifs and temporal network representations, Appl. Network Sci. 7 (2022), Paper No. 4. 10.1007/s41109-021-00441-zSuche in Google Scholar
[3] S. Chowdhury and F. Mémoli, Persistent path homology of directed networks, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia (2018), 1152–1169. 10.1137/1.9781611975031.75Suche in Google Scholar
[4] A. Grigor’yan, R. Jimenez, Y. V. Muranov and S.-T. Yau, On the path homology theory of digraphs and Eilenberg–Steenrod axioms, Homology Homotopy Appl. 20 (2018), no. 2, 179–205. 10.4310/HHA.2018.v20.n2.a9Suche in Google Scholar
[5] A. Grigor’yan, Y. Lin, Y. V. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, preprint (2012), https://arxiv.org/abs/1207.2834. Suche in Google Scholar
[6] A. Grigor’yan, Y. Lin, Y. V. Muranov and S.-T. Yau, Homotopy theory for digraphs, Pure Appl. Math. Q. 10 (2014), no. 4, 619–674. 10.4310/PAMQ.2014.v10.n4.a2Suche in Google Scholar
[7] A. Grigor’yan, Y. Lin, Y. V. Muranov and S.-T. Yau, Cohomology of digraphs and (undirected) graphs, Asian J. Math. 19 (2015), no. 5, 887–932. 10.4310/AJM.2015.v19.n5.a5Suche in Google Scholar
[8] A. Grigor’yan, Y. Lin, Y. V. Muranov and S.-T. Yau, Path complexes and their homologies, J. Math. Sci. 248 (2020), 564–599. 10.1007/s10958-020-04897-9Suche in Google Scholar
[9] A. Grigor’yan, Y. V. Muranov, V. Vershinin and S.-T. Yau, Path homology theory of multigraphs and quivers, Forum Math. 30 (2018), no. 5, 1319–1337. 10.1515/forum-2018-0015Suche in Google Scholar
[10] A. Grigor’yan, Y. V. Muranov and S.-T. Yau, Graphs associated with simplicial complexes, Homology Homotopy Appl. 16 (2014), no. 1, 295–311. 10.4310/HHA.2014.v16.n1.a16Suche in Google Scholar
[11] A. Grigor’yan, Y. V. Muranov and S.-T. Yau, Homologies of digraphs and Künneth formulas, Comm. Anal. Geom. 25 (2017), no. 5, 969–1018. 10.4310/CAG.2017.v25.n5.a4Suche in Google Scholar
[12] C. A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge University, Cambridge, 1995. Suche in Google Scholar
© 2026 Walter de Gruyter GmbH, Berlin/Boston
