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Hyperbolicity in the corona and join of graphs


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 GH and the corona \({G \odot \mathcal H}\).

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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).

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Mathematics Subject Classification

  • 05C69
  • 05A20
  • 05C50


  • Graph join
  • Corona graph
  • Gromov hyperbolicity
  • Infinite graph