Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Hyperbolicity in the corona and join of graphs

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 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 \({G \odot \mathcal H}\).

This is a preview of subscription content, log in to check access.

References

  1. 1

    Alonso, J., Brady, T., Cooper, D., Delzant, T., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M., Short, H.: Notes on word hyperbolic groups. In: Ghys, E., Haefliger, A., Verjovsky, A. Group Theory from a Geometrical Viewpoint, pp. . World Scientific, Singapore (1992)

  2. 2

    Bermudo, S., Rodríguez, J.M., Rosario, O., Sigarreta, J.M.: Small values of the hyperbolicity constant in graphs (submitted, preprint). http://gama.uc3m.es/index.php/jomaro.html

  3. 3

    Bermudo S., Rodríguez J.M., Sigarreta J.M.: Computing the hyperbolicity constant. Comput. Math. Appl. 62, 4592–4595 (2011)

  4. 4

    Bermudo S., Rodríguez J.M., Sigarreta J.M., Vilaire J.-M.: Gromov hyperbolic graphs. Discret Math. 313, 1575–1585 (2013)

  5. 5

    Bermudo S., Rodríguez J.M., Sigarreta J.M., Tourís E.: Hyperbolicity and complement of graphs. Appl. Math. Lett. 24, 1882–1887 (2011)

  6. 6

    Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. In: Ghys, E., Haefliger, A., Verjovsky, A. (eds.) Group Theory from a Geometrical Viewpoint, Trieste, 1990. World Scientific, River Edge, pp. 64–167 (1991)

  7. 7

    Brinkmann G., Koolen J., Moulton V.: On the hyperbolicity of chordal graphs. Ann. Comb. 5, 61–69 (2001)

  8. 8

    Carballosa W., Casablanca R.M., de la Cruz A., Rodríguez J.M.: Gromov hyperbolicity in strong product graphs. Electr. J. Comb. 20(3), P2 (2013)

  9. 9

    Carballosa, W., de la Cruz, A., Rodríguez, J.M.: Gromov hyperbolicity in lexicographic product graphs (submitted, preprint). http://gama.uc3m.es/index.php/jomaro.html

  10. 10

    Carballosa W., Rodríguez J.M., Sigarreta J.M., Villeta M.: Gromov hyperbolicity of line graphs. Electr. J. Comb. 18, P210 (2011)

  11. 11

    Charney R.: Artin groups of finite type are biautomatic. Math. Ann. 292, 671–683 (1992)

  12. 12

    Chen B., Yau S.-T., Yeh Y.-N.: Graph homotopy and Graham homotopy. Discrete Math. 241, 153–170 (2001)

  13. 13

    Chepoi, V., Estellon, B.: Packing and covering đ-hyperbolic spaces by balls, APPROX-RANDOM, pp. 59–73 (2007)

  14. 14

    Chia G.L.: On the join of graph and chromatic uniqueness. J. Graph Theory 19(2), 251–261 (2005)

  15. 15

    Eppstein, D.: Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition, SODA’2007. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 29-38 (2007)

  16. 16

    Frucht R., Harary F.: On the corona of two graphs. Aequ. Math. 4(3), 322–324 (1970)

  17. 17

    Gavoille, C., Ly, O.: Distance labeling in hyperbolic graphs. In: ISAAC, pp. 171–179 (2005)

  18. 18

    Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. In: Progress in Mathematics, vol. 83. Birkhäuser Boston Inc., Boston (1990)

  19. 19

    Gromov, M.: Hyperbolic groups, in “Essays in group theory”. In: Gersten, S.M. (ed.) M.S.R.I. Publications, vol. 8. Springer, New York, pp. 75–263 (1987)

  20. 20

    Harary F.: Graph Theory. Addison-Wesley, Reading (1994)

  21. 21

    Jonckheere, E., Lohsoonthorn, P.: A hyperbolic geometry approach to multipath routing. In: Proceedings of the 10th Mediterranean Conference on Control and Automation (MED 2002), Lisbon, FA5-1 (2002)

  22. 22

    Jonckheere E.A.: Contrôle du traffic sur les réseaux à géométrie hyperbolique–Vers une théorie géométrique de la sécurité l’acheminement de l’information. J. Eur. Syst. Autom. 8, 45–60 (2002)

  23. 23

    Jonckheere E.A., Lohsoonthorn P.: Geometry of network security. Am. Control Conf. ACC, 111–151 (2004)

  24. 24

    Klešč M.: The join of graph and crossing number. Electron. Note Discrete Math. 28, 349–355 (2007)

  25. 25

    Krauthgamer, R., Lee, J.R.: Algorithms on negatively curved spaces. In: Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 119–128, New York (2006). doi:10.1109/FOCS.2006.9

  26. 26

    Krioukov D., Papadopoulos F., Kitsak M., Vahdat A., Boguñá M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)

  27. 27

    Kuziak D., Yero I.G., Rodríguez-Velazquez J.A.: On the strong metric dimension of corona product graphs and join graphs. Discrete Appl. Math. 161(7), 1022–1027 (2013)

  28. 28

    Michel J., Rodríguez J.M., Sigarreta J.M., Villeta M.: Hyperbolicity and parameters of graphs. ARS Comb. 100, 43–63 (2011)

  29. 29

    Michel J., Rodríguez J.M., Sigarreta J.M., Villeta M.: Gromov hyperbolicity in cartesian product graphs. Proc. Indian Acad. Sci. Math. Sci. 120, 1–17 (2010)

  30. 30

    Narayan O., Saniee I.: Large-scale curvature of networks. Phys. Rev. E 84, 066108 (2011)

  31. 31

    Oshika, K.: Discrete groups. Iwanami Series in Modern Mathematics, vol. 207, pp. 1–193, American Mathematical Society, New York. http://www.ams.org/bookstore?fn=20&arg1=mmonoseries&ikey=MMONO-207 (2002)

  32. 32

    Pestana D., Rodríguez J.M., Sigarreta J.M., Villeta M.: Gromov hyperbolic cubic graphs. Central Eur. J. Math. 10(3), 1141–1151 (2012)

  33. 33

    Portilla, A., Rodríguez, J.M., Sigarreta, J.M., Vilaire, J.-M.: Gromov hyperbolic tessellation graphs. Utilitas Math. (to appear, preprint). http://gama.uc3m.es/index.php/jomaro.html

  34. 34

    Rodríguez J.M.: Characterization of Gromov hyperbolic short graphs. Acta Math. Sin. 30, 197–212 (2014)

  35. 35

    Rodríguez J.M., Sigarreta J.M., Vilaire J.-M., Villeta M.: On the hyperbolicity constant in graphs. Discrete Math. 311, 211–219 (2011)

  36. 36

    Shavitt, Y., Tankel, T.: On internet embedding in hyperbolic spaces for overlay construction and distance estimation. In: INFOCOM (2004)

  37. 37

    Sigarreta, J.M.: Hyperbolicity in median graphs. Proc. Indian Acad. Sci. Math. Sci. (2013) (in press)

  38. 38

    Yero I.G., Kuziak D., A. R.: Aguilar, coloring, location and domination of corona graphs. Aequ. Math. 86(1–2), 1–21 (2013)

Download references

Author information

Correspondence to Walter Carballosa.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carballosa, W., Rodríguez, J.M. & Sigarreta, J.M. Hyperbolicity in the corona and join of graphs. Aequat. Math. 89, 1311–1327 (2015). https://doi.org/10.1007/s00010-014-0324-0

Download citation

Mathematics Subject Classification

  • 05C69
  • 05A20
  • 05C50

Keywords

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