# Gromov Hyperbolicity in Directed Graphs

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

**Notation.**For each geodesic metric space X, we shall denote by ${d}_{X}$ and ${L}_{X}$, the distance and the length, respectively.

## 2. Background on Hyperbolic Spaces

**Definition**

**1.**

**Definition**

**2.**

**Example**

**1.**

**Definition**

**3.**

**Theorem**

**1.**

**Definition**

**4.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. Directed Graphs

**Definition**

**5.**

**Definition**

**6.**

**Remark**

**1.**

**Definition**

**7.**

**Definition**

**8.**

- I
- For every $a\in A$ there exists $b\in B$ such that ${D}_{G}(a,b)\u2a7d\alpha $ and ${D}_{G}(b,a)\u2a7d\alpha $.
- II
- For every $b\in B$ there exists $a\in A$ such that ${D}_{G}(b,a)\u2a7d\alpha $ and ${D}_{G}(a,b)\u2a7d\alpha $.

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

- I
- For each $\delta ,b,D\u2a7e0$ and $a,C\u2a7e1$, there exists a constant $H=H(\delta ,a,b,C,D)$, which only depends on these five parameters, with the following property:Let us consider any admissible δ-hyperbolic directed graph G with $C\left(G\right)\u2a7dC$ and $D\left(G\right)\u2a7dD$, any $x,y\in G,$ and any $(a,b)$-quasigeodesic g starting at x and ending at y. If γ is any geodesic starting at x and ending at y (or starting at y and ending at x), then ${\mathcal{H}}_{G}(g,\gamma )\u2a7dH$.
- II
- Let us consider any admissible directed graph G with $C\left(G\right),D\left(G\right)$ finite numbers, satisfying the following property.For each $a\u2a7e1$ and $b\u2a7e0$ there exists a constant H such that for any $x,y\in G,$ any $(a,b)$-quasigeodesic g in G starting at x and ending at y, and any geodesic γ starting at x and ending at y, the inequality ${\mathcal{H}}_{G}(g,\gamma )\u2a7dH$ holds.Then, G is hyperbolic.

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gromov, M. Hyperbolic Groups, in Essays in Group Theory; Springer: Berlin, Germany, 1987; pp. 75–263. [Google Scholar]
- Ghys, E.; de la Harpe, P. Sur les Groupes Hyperboliques d’après Mikhael Gromov; Birkhäuser: Berlin, Germany, 1990. [Google Scholar]
- Bonk, M.; Schramm, O. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal.
**2000**, 10, 266–306. [Google Scholar] [CrossRef] - Foertsch, T.; Schroeder, V. A product construction for hyperbolic metric spaces. Ill. J. Math.
**2005**, 49, 793–810. [Google Scholar] [CrossRef] - Naor, A.; Peres, Y.; Schramm, O.; Sheffield, S. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J.
**2006**, 134, 165–197. [Google Scholar] [CrossRef] [Green Version] - Wenger, S. Gromov hyperbolic spaces and the sharp isoperimetric constant. Invent. Math.
**2008**, 171, 227–255. [Google Scholar] [CrossRef] [Green Version] - Aramayona, J. Simplicial embeddings between pants graphs. Geom Dedicata
**2010**, 144, 115–128. [Google Scholar] [CrossRef] [Green Version] - Holopainen, I.; Soardi, P.M. p-harmonic functions on graphs and manifolds. Manuscripta Math.
**1997**, 94, 95–110. [Google Scholar] [CrossRef] - Kanai, M. Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds. J. Math. Soc. Jpn.
**1985**, 37, 391–413. [Google Scholar] [CrossRef] - Portilla, A.; Rodríguez, J.M.; Tourís, E. Stability of Gromov hyperbolicity. J. Adv. Math. Stud.
**2009**, 2, 1–20. [Google Scholar] - Portilla, A.; Tourís, E. A characterization of Gromov hyperbolicity of surfaces with variable negative curvature. Publ. Mat.
**2009**, 53, 83–110. [Google Scholar] [CrossRef] [Green Version] - Bowditch, B.H. Notes on Gromov’s Hyperobolicity Criterion for Path-Metric Spaces. In Group Theory from a Geometrical Viewpoint; World Scientific: River Edge, NJ, USA, 1991; pp. 64–167. [Google Scholar]
- Portilla, A.; Rodríguez, J.M.; Tourís, E. Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal.
**2004**, 14, 123–149. [Google Scholar] [CrossRef] [Green Version] - Rodríguez, J.M.; Tourís, E. Gromov hyperbolicity of Riemann surfaces. Acta Math. Sin.
**2007**, 23, 209–228. [Google Scholar] [CrossRef] [Green Version] - Tourís, E. Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces. J. Math. Anal. Appl.
**2011**, 380, 865–881. [Google Scholar] [CrossRef] [Green Version] - Bermudo, S.; Carballosa, W.; Rodríguez, J.M.; Sigarreta, J.M. On the hyperbolicity of edge-chordal and path-chordal graphs. Filomat
**2016**, 30, 2599–2607. [Google Scholar] [CrossRef] - Bermudo, S.; Rodríguez, J.M.; Sigarreta, J.M. Computing the hyperbolicity constant. Comput. Math. Appl.
**2011**, 62, 4592–4595. [Google Scholar] [CrossRef] [Green Version] - Bermudo, S.; Rodríguez, J.M.; Sigarreta, J.M.; Touris, E. Hyperbolicity and complement of graphs. Appl. Math. Lett.
**2011**, 24, 1882–1887. [Google Scholar] [CrossRef] [Green Version] - Brinkmann, G.; Koolen, J.; Moulton, V. On the hyperbolicity of chodal graphs. Ann. Comb.
**2001**, 5, 61–69. [Google Scholar] [CrossRef] - Cantón, A.; Granados, A.; Pestana, P.; Rodríguez, J.M. Gromov hyperbolicity of periodic graphs. Bull. Malays. Math. Sci. Soc.
**2016**, 39, S89–S116. [Google Scholar] [CrossRef] - Carballosa, W.; de la Cruz, A.; Martínez-Pŕez, A.; Rodríguez, J.M. Hyperbolicity of direct products of graphs. Symmetry
**2018**, 10, 279. [Google Scholar] [CrossRef] [Green Version] - Frigerio, R.; Sisto, A. Characterizing hyperbolic spaces and real trees. Geom. Dedicata
**2009**, 142, 139–149. [Google Scholar] [CrossRef] [Green Version] - Gavoille, G.; Ly, O. Distance Labeling in Hyperbolic Graphs; Springer: Berlin, Germany, 2005; pp. 1071–1079. [Google Scholar]
- Granados, A.; Pestana, D.; Portilla, A.; Rodríguez, J.M. Gromov hyperbolicity in Mycielskian Graphs. Symmetry
**2017**, 9, 131. [Google Scholar] [CrossRef] [Green Version] - Grippo, E.; Jonckheere, E.A. Effective resistance criterion for negative curvature: Application to congestion control. In Proceedings of the 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, Argentina, 19–22 September 2016. [Google Scholar]
- Hernández, J.C.; Reyes, R.; Rodríguez, J.M.; Sigarreta, J.M. Mathematical properties on the hyperbolicity of interval graphs. Symmetry
**2017**, 9, 255. [Google Scholar] [CrossRef] [Green Version] - Jonckheere, E.A.; Lohsoonthorn, P. Geometry of Network Security. In Proceedings of the 2004 American Control Conference, Boston, MA, USA, 30 June–2 July 2004; IEEE: Piscataway, NJ, USA; pp. 111–151. [Google Scholar] [CrossRef]
- Jonckheere, E.; Lohsoonthorn, P.; Bonahon, F. Scaled Gromov hyperbolic graphs. J. Graph Theory
**2008**, 57, 157–180. [Google Scholar] [CrossRef] - Koolen, J.H.; Moulton, V. Hyperbolic Bridged Graphs. Eur. J. Comb.
**2002**, 23, 683–699. [Google Scholar] [CrossRef] - Méndez-Bermúdez, J.A.; Reyes, R.; Rodríguez, J.M.; Sigarreta, J.M. Hyperbolicity on graph operators. Symmetry
**2018**, 10, 360. [Google Scholar] [CrossRef] [Green Version] - Rodríguez, J.M.; Sigarreta, J.M.; Vilaire, J.-M.; Villeta, M. On the hyperbolicity constant in graphs. Discret. Math.
**2011**, 311, 211–219. [Google Scholar] [CrossRef] [Green Version] - Shang, Y. Lack of Gromov-hyperbolicity in colored random networks. Pan Am. Math. J.
**2011**, 21, 27–36. [Google Scholar] [CrossRef] - Shang, Y. Lack of Gromov-hyperbolicity in small-world networks. Cent. Eur. J. Math.
**2012**, 10, 1152–1158. [Google Scholar] [CrossRef] - Shang, Y. Random Lifts of Graphs: Network Robustness Based on The Estrada Index. Appl. Math. E Notes
**2012**, 12, 53–61. [Google Scholar] - Shang, Y. Non-hyperbolicity of random graphs with given expected degrees. Stoch. Model.
**2013**, 29, 451–462. [Google Scholar] [CrossRef] - Shang, Y. On the likelihood of forests. Phys. A Stat. Mech. Appl.
**2016**, 456, 157–166. [Google Scholar] [CrossRef] - Wu, Y.; Zhang, C. Hyperbolicity and chordality of a graph. Electr. J. Combin.
**2011**, 18, 43. [Google Scholar] - Charney, R. Artin groups of finite type are biautomatic. Math. Ann.
**1992**, 292, 671–683. [Google Scholar] [CrossRef] - Baryshnikov, Y. On the Curvature of the Internet; Workshop on Stochastic Geometry and Teletraffic: Eindhoven, The Netherlands, 2002. [Google Scholar]
- Jonckheere, E.; Lohsoonthorn, P.; Ariaei, F. Upper bound on scaled Gromov hyperbolic delta. Appl. Math. Comput.
**2007**, 192, 191–204. [Google Scholar] [CrossRef] - Kleinberg, R. Geographic routing using hyperbolic space. In Proceedings of the IEEE INFOCOM 2007—26th IEEE International Conference on Computer Communications, Barcelona, Spain, 6–12 May 2007; IEEE: Piscataway, NJ, USA, 2007; pp. 1902–1909. [Google Scholar] [CrossRef] [Green Version]
- Jonckheere, E.A. Contrôle du trafic sur les réseaux à géometrie hyperbolique–Une approche mathématique a la sécurité de l’acheminement de l’information. J. Eur. Syst. Autom.
**2003**, 37, 145–159. [Google Scholar] - Hespanha, J.P.; Bohacek, S. Preliminary Results in Routing Games. In Proceedings of the 2001 American Control Conference, Arlington, VA, USA, 25–27 June 2001. [Google Scholar] [CrossRef]
- Jonckheere, E.A.; Lohsoonthorn, P. A hyperbolic geometry approach to multi-path routing. In Proceedings of the 10th Mediterranean Conference on Control and Automation (MED 2002), Lisbon, Portugal, 9–12 July 2002. FA5-1. [Google Scholar]
- Anderson, J.W. Hyperbolic Geometry; Springer: London, UK, 1999. [Google Scholar]
- Chen, B.; Yau, S.-T.; Yeh, Y.-N. Graph homotopy and Graham homotopy. Discret. Math.
**2001**, 241, 153–170. [Google Scholar] [CrossRef] [Green Version] - Bonk, M. Quasi-geodesics segments and Gromov hyperbolic spaces. Geom. Dedicata
**1996**, 62, 281–298. [Google Scholar] [CrossRef] [Green Version] - Hausdorff, F. Set Theory; American Mathematical Society (English translation): Providence, RI, USA, 1957. [Google Scholar]
- Künzi, H.P.A. Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology. In Handbook of the History of General Topology; Kluwer Academic Publishers: Boston, MA, USA, 2001; Volume 3, pp. 853–968. [Google Scholar]
- Schellekens, M. The Smyth completion: A common foundation for denotational semantics and complexity analysis. Electr. Notes Theor. Comp. Sci.
**1995**, 1, 211–232. [Google Scholar] [CrossRef] [Green Version] - García-Raffi, L.M.; Romaguera, S.; Sánchez-Pérez, E.A. Sequence spaces and asymmetric norms in the theory of computational complexity. Math. Comp. Modell.
**2002**, 36, 1–11. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) An admissible graph G. (In bold face, the directed geodesic joining A and E); (

**b**) Its canonical graph ${G}_{0}$. (In bold face, the geodesic joining A and E).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Portilla, A.; Rodríguez, J.M.; Sigarreta, J.M.; Tourís, E.
Gromov Hyperbolicity in Directed Graphs. *Symmetry* **2020**, *12*, 105.
https://doi.org/10.3390/sym12010105

**AMA Style**

Portilla A, Rodríguez JM, Sigarreta JM, Tourís E.
Gromov Hyperbolicity in Directed Graphs. *Symmetry*. 2020; 12(1):105.
https://doi.org/10.3390/sym12010105

**Chicago/Turabian Style**

Portilla, Ana, José M. Rodríguez, José M. Sigarreta, and Eva Tourís.
2020. "Gromov Hyperbolicity in Directed Graphs" *Symmetry* 12, no. 1: 105.
https://doi.org/10.3390/sym12010105