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Hyperbolicity in the corona and join of graphs

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Abstract

If X is a geodesic metric space and \({x_1, x_2, x_3 \in X}\), 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 the 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. δ(X) = inf{δ ≥ 0: X is δ-hyperbolic}. In this paper we characterize the hyperbolic product graphs for graph join G + H and the corona \({G\odot\mathcal H: G + H}\) is always hyperbolic, and \({G\odot\mathcal H}\) is hyperbolic if and only if G is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join G +  H and the corona \({G \odot \mathcal H}\).

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Authors and Affiliations

  1. Consejo Nacional de Ciencia y Tecnología (CONACYT) and Universidad Autónoma de Zacatecas, Paseo la Bufa, int. Calzada Solidaridad, 98060, Zacatecas, ZAC, Mexico

    Walter Carballosa

  2. Department of Mathematics, Universidad Carlos III de Madrid, Av. de la Universidad 30, Leganés, 28911, Madrid, Spain

    José M. Rodríguez

  3. Faculdad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico

    José M. Sigarreta

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  1. Walter Carballosa
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  3. José M. Sigarreta
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Correspondence to Walter Carballosa.

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Carballosa, W., Rodríguez, J.M. & Sigarreta, J.M. Hyperbolicity in the corona and join of graphs. Aequat. Math. 89, 1311–1327 (2015). https://doi.org/10.1007/s00010-014-0324-0

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  • DOI: https://doi.org/10.1007/s00010-014-0324-0

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