Scaled Gromov hyperbolic graphs

  1. Edmond Jonckheere1,
  2. Poonsuk Lohsoonthorn1,
  3. Francis Bonahon2

Article first published online: 12 OCT 2007

DOI: 10.1002/jgt.20275

Issue

Journal of Graph Theory

Journal of Graph Theory

Volume 57, Issue 2, pages 157–180, February 2008

Additional Information

How to Cite

Jonckheere, E., Lohsoonthorn, P. and Bonahon, F. (2008), Scaled Gromov hyperbolic graphs. Journal of Graph Theory, 57: 157–180. doi: 10.1002/jgt.20275

Author Information

  1. 1

    Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089-2563

  2. 2

    Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532

Publication History

  1. Issue published online: 11 DEC 2007
  2. Article first published online: 12 OCT 2007
  3. Manuscript Revised: 20 AUG 2007
  4. Manuscript Received: 19 OCT 2005

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Keywords:

  • Gromov hyperbolic graph;
  • CAT (Cartan–Alexandrov–Toponogov) space;
  • comparison theory;
  • Higuchi local curvature;
  • scale-free network

Abstract

In this article, the δ-hyperbolic concept, originally developed for infinite graphs, is adapted to very large but finite graphs. Such graphs can indeed exhibit properties typical of negatively curved spaces, yet the traditional δ-hyperbolic concept, which requires existence of an upper bound on the fatness δ of the geodesic triangles, is unable to capture those properties, as any finite graph has finite δ. Here the idea is to scale δ relative to the diameter of the geodesic triangles and use the Cartan–Alexandrov–Toponogov (CAT) theory to derive the thresholding value of δdiam below which the geometry has negative curvature properties. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 157–180, 2008

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