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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.35 no.2 Antofagasta jun. 2016

http://dx.doi.org/10.4067/S0716-09172016000200001 

Proyecciones Journal of Mathematics Vol. 35, No 2, pp. 137-157, June 2016. Universidad Católica del Norte Antofagasta - Chile

Closed models, strongly connected components and Euler graphs

Tsemo Aristide

College Boreal, Canadá 


ABSTRACT

In this paper, we continue our study of closed models defined in categories of graphs. We construct a closed model defined in the cat-egory 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).

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[6]    Cisinski D. C. Les préfaisceaux comme type d’homotopie, Asterisque, Volume 308, Soc. Math. France, (2006).

[7]    Euler, L. Solutio problematis ad geometriam situs pertinentis. Com-mentarii academiae scientiarum Petropolitanae, 8, pp. 128-140, (1741).

[8]    Artin, M., Grothendieck, A., Verdier, J. L. Theorie des topos et coho-mologie 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.

Tsemo Aristide

College Boreal,

1 Yonge Street,

M5E 1E5, Toronto, ON Canada

e-mail : tsemo58@yahoo.ca

Received : January 2015. Accepted : March 2016

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