Bounds on Gromov hyperbolicity constant

  • Verónica Hernández
  • Domingo Pestana
  • José M. Rodríguez
Original Paper


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 \(\delta \)-hyperbolic in the Gromov sense if any side of T is contained in a \(\delta \)-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by \(\delta (X)\) the sharp hyperbolicity constant of X, i.e., \(\delta (X) =\inf \{ \delta \ge 0: X ~\text {is}~ \delta \text {-hyperbolic} \}.\) To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolicity constant in terms of some parameters of the graph. Denote by \(\mathcal {G}(n,m)\) the set of (simple) graphs G with n vertices and m edges, and such that every edge has length 1. In this work we estimate \(A(n,m):=\min \{\delta (G)\mid G \in \mathcal {G}(n,m) \}\) and \(B(n,m):=\max \{\delta (G)\mid G \in \mathcal {G}(n,m) \}\). In particular, we obtain good bounds for B(nm), and we compute the precise value of A(nm) for all values of n and m. We also study this problem for non-simple and weighted graphs.


Gromov hyperbolicity Hyperbolicity constant Finite graphs Geodesic 

Mathematics Subject Classification

05C75 05C12 05A20 05C80 



This work was partially supported by a Grant from Ministerio de Economía y Competitividad (MTM 2013-46374-P), Spain. We would like to thank the referees for their careful reading of the manuscript and several useful comments which have helped us to improve the paper.


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Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  • Verónica Hernández
    • 1
  • Domingo Pestana
    • 1
  • José M. Rodríguez
    • 1
  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

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