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Bounds on Gromov hyperbolicity constant in graphs

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Abstract

If X is a geodesic metric space and x 1,x 2,x 3 ∈ X, a geodesic triangle T = {x 1,x 2,x 3} is the union of the three geodesics [x 1 x 2], [x 2 x 3] and [x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. \(\delta(X)=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\}\,. \) In this paper we relate the hyperbolicity constant of a graph with some known parameters of the graph, as its independence number, its maximum and minimum degree and its domination number. Furthermore, we compute explicitly the hyperbolicity constant of some class of product graphs.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    JOSÉ M RODRÍGUEZ

  2. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, 39650, Acalpulco Gro., Mexico

    JOSÉ M SIGARRETA

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Correspondence to JOSÉ M RODRÍGUEZ.

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RODRÍGUEZ, J.M., SIGARRETA, J.M. Bounds on Gromov hyperbolicity constant in graphs. Proc Math Sci 122, 53–65 (2012). https://doi.org/10.1007/s12044-012-0060-0

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  • DOI: https://doi.org/10.1007/s12044-012-0060-0

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