Aequationes mathematicae

, Volume 89, Issue 5, pp 1311–1327 | Cite as

Hyperbolicity in the corona and join of graphs

  • Walter Carballosa
  • José M. Rodríguez
  • José M. Sigarreta
Article

Abstract

If X is a geodesic metric space and \({x_1, x_2, x_3 \in X}\), a geodesic triangleT = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] 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 \({G \odot \mathcal H}\).

Mathematics Subject Classification

05C69 05A20 05C50 

Keywords

Graph join Corona graph Gromov hyperbolicity Infinite graph 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Walter Carballosa
    • 1
  • José M. Rodríguez
    • 2
  • José M. Sigarreta
    • 3
  1. 1.Consejo Nacional de Ciencia y Tecnología (CONACYT) and Universidad Autónoma de ZacatecasZacatecasMexico
  2. 2.Department of MathematicsUniversidad Carlos III de MadridLeganésSpain
  3. 3.Faculdad de MatemáticasUniversidad Autónoma de GuerreroGuerreroMexico

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