Discrete cubical and path homologies of graphs
Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 417-437.

In this paper we study and compare two homology theories for (simple and undirected) graphs. The first, which was developed by Barcelo, Capraro, and White, is based on graph maps from hypercubes to the graph. The second theory was developed by Grigor’yan, Lin, Muranov, and Yau, and is based on paths in the graph. Results in both settings imply that the respective homology groups are isomorphic in homological dimension one. We show that, for several infinite classes of graphs, the two theories lead to isomorphic homology groups in all dimensions. However, we provide an example for which the homology groups of the two theories are not isomorphic at least in dimensions two and three. We establish a natural map from the cubical to the path homology groups which is an isomorphism in dimension one and surjective in dimension two. Again our example shows that in general the map is not surjective in dimension three and not injective in dimension two. In the process we develop tools to compute the homology groups for both theories in all dimensions.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.49
Classification: 05C99, 55U99
Keywords: Discrete cubical homology, path homology, homology of graphs
Barcelo, Hélène 1; Greene, Curtis 2; Jarrah, Abdul Salam 3; Welker, Volkmar 4

1 The Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720, USA
2 Haverford College, Haverford, PA 19041, USA
3 Department of Mathematics and Statistics, American University of Sharjah, PO Box 26666, Sharjah, United Arab Emirates
4 Fachbereich Mathematik und Informatik Philipps-Universität 35032 Marburg, Germany
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Barcelo, Hélène; Greene, Curtis; Jarrah, Abdul Salam; Welker, Volkmar. Discrete cubical and path homologies of graphs. Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 417-437. doi : 10.5802/alco.49. http://www.numdam.org/articles/10.5802/alco.49/`

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