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

Keywords: Discrete cubical homology, path homology, homology of graphs

^{1}; Greene, Curtis

^{2}; Jarrah, Abdul Salam

^{3}; Welker, Volkmar

^{4}

@article{ALCO_2019__2_3_417_0, author = {Barcelo, H\'el\`ene and Greene, Curtis and Jarrah, Abdul Salam and Welker, Volkmar}, title = {Discrete cubical and path homologies of graphs}, journal = {Algebraic Combinatorics}, pages = {417--437}, publisher = {MathOA foundation}, volume = {2}, number = {3}, year = {2019}, doi = {10.5802/alco.49}, zbl = {07066882}, mrnumber = {3968745}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.49/} }

TY - JOUR AU - Barcelo, Hélène AU - Greene, Curtis AU - Jarrah, Abdul Salam AU - Welker, Volkmar TI - Discrete cubical and path homologies of graphs JO - Algebraic Combinatorics PY - 2019 SP - 417 EP - 437 VL - 2 IS - 3 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.49/ DO - 10.5802/alco.49 LA - en ID - ALCO_2019__2_3_417_0 ER -

%0 Journal Article %A Barcelo, Hélène %A Greene, Curtis %A Jarrah, Abdul Salam %A Welker, Volkmar %T Discrete cubical and path homologies of graphs %J Algebraic Combinatorics %D 2019 %P 417-437 %V 2 %N 3 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.49/ %R 10.5802/alco.49 %G en %F ALCO_2019__2_3_417_0

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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