# Bounds on Gromov hyperbolicity constant

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

## 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 $$\delta$$-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.

## Keywords

Gromov hyperbolicity Hyperbolicity constant Finite graphs Geodesic

## Mathematics Subject Classification

05C75 05C12 05A20 05C80

## Notes

### Acknowledgments

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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