Closed models, strongly connected components and Euler graphs
DOI:
https://doi.org/10.4067/S0716-09172016000200001Keywords:
Euler graphs, closed models, grafos de Euler, modelos cerrados.Abstract
In this paper, we continue our study of closed models defined in categories of graphs. We construct a closed model defined in the category of directed graphs which characterizes the strongly connected components. This last notion has many applications, and it plays an important role in the web search algorithm of Brin and Page, the foun-dation of the search engine Google. We also show that for this closed model, Euler graphs are particular examples of cofibrant objects. This enables us to interpret in this setting the classical result of Euler which states that a directed graph is Euleurian if and only if the in degree and the out degree of every of its nodes are equal. We also provide a cohomological proof of this last result.References
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[2] Bisson, T., Tsemo, A. A homotopical algebra of graphs related to zeta series. Homology, Homotopy and Applications, 11 (1), pp. 171-184, (2009).
[3] Bisson, T., Tsemo, A. Symbolic dynamics and the category of graphs. Theory and Applications of Categories, 25 (22), pp. 614-640, (2011).
[4] Bisson, T., Tsemo, A. Homotopy equivalence of isospectral graphs. New York J. Math, 17, pp. 295-320, (2011).
[5] Brin, S., Page, L. Reprint of: The anatomy of a large-scale hypertextual web search engine. Computer networks, 56 (18), pp. 3825-3833, (2012).
[6] Cisinski D. C. Les pr´ efaisceaux comme type d’homotopie, Asterisque, Volume 308, Soc. Math. France, (2006).
[7] Euler, L. Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, 8, pp. 128-140, (1741).
[8] Artin, M., Grothendieck, A., Verdier, J. L. Theorie des topos et cohomologie etale des schemas. Tome 1. Lecture notes in mathematics, 269, (1972).
[9] Hirschhorn, P. S. Model categories and their localizations (No. 99). American Mathematical Soc., (2009).
[10] Tsemo, A. (2013). Applications of closed models defined by counting to graph theory and topology. arXiv preprint arXiv:1308.3983
Published
2017-03-23
How to Cite
[1]
T. Aristide, “Closed models, strongly connected components and Euler graphs”, Proyecciones (Antofagasta, On line), vol. 35, no. 2, pp. 137-157, Mar. 2017.
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