Abstract
Higher homotopy of graphs has been defined in several articles. In Dochterman (Hom complexes and homotopy theory in the category of graphs. arXiv math/0605275 v2,28/09/2006, 2006), the authors asked for a companion homology theory. We define such a theory for the category of unoriented reflexive graphs; it exhibits a long exact sequence for a pair of graphs (G, A), satisfies an excision property and a Hurewicz theorem. This allows us to compute the top homology of the graphical n-spheres showing that the theory is not trivial and is able to detect n-dimensional holes in a graph. The long-term objective is to compare the homotopy of the topological and graphical spheres.
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Talbi, M.E., Benayat, D. Homology Theory of Graphs. Mediterr. J. Math. 11, 813–828 (2014). https://doi.org/10.1007/s00009-013-0358-x
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DOI: https://doi.org/10.1007/s00009-013-0358-x