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Advances in Mathematical Physics
Volume 2015, Article ID 723451, 11 pages
http://dx.doi.org/10.1155/2015/723451
Research Article

Mathematical Properties of the Hyperbolicity of Circulant Networks

1Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54, Colonia Garita, 39650 Acapulco, GRO, Mexico
2Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, 28911 Madrid, Spain

Received 24 July 2015; Accepted 27 September 2015

Academic Editor: Pavel Kurasov

Copyright © 2015 Juan C. Hernández et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

If is a geodesic metric space and , a geodesic triangle   is the union of the three geodesics , , and in . The space is -hyperbolic (in the Gromov sense) if any side of is contained in a -neighborhood of the union of the two other sides, for every geodesic triangle in . The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.