Expanders are not hyperbolic. (English) Zbl 0915.05072
Summary: We bound from above the number of vertices of a graph in terms of the Cheeger constant and the \(\delta\)-hyperbolicity of the graph. As a corollary we get that expanders are not uniformly hyperbolic.
MSC:
| 05C35 | Extremal problems in graph theory |
Keywords:
hyperbolic graphs; Cayley graphs; hyperbolicity constant; bound; Cheeger constant; expanders; uniformly hyperbolicReferences:
| [1] | A. Ancona,Positive harmonic functions and hyperbolicity, Lecture Notes in Mathematics1344, Springer-Verlag, Berlin, 1988, pp. 1–23. |
| [2] | E Ghys, A. Hafliger and A. Verjovsky,Group Theory from a Geometrical Viewpoint, World Scientific, Singapore, 1991. |
| [3] | M. Gromov,Hyperbolic groups, Mathematical Sciences Research Institute Publications8 (1987), 75–263. · Zbl 0634.20015 |
| [4] | A. Lubotzky,Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser, Boston, 1994. · Zbl 0826.22012 |
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