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Geometric & Functional Analysis GAFA

, Volume 7, Issue 3, pp 403–419 | Cite as

Every Graph with a Positive Cheeger Constant Contains a Tree with a Positive Cheeger Constant

  • I. Benjamini
  • O. Schramm

Abstract.

It is shown that every (infinite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the Cheeger constant. This minimal graph, T(k), is a tree, and every graph G with Cheeger constant \( h(G) \geq k \) has a spanning forest in which each component is isomorphic to T(k).

Keywords

Nonnegative Integer Connected Graph Minimal Graph Span Forest Cheeger Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 1997

Authors and Affiliations

  • I. Benjamini
    • 1
  • O. Schramm
    • 1
  1. 1.Itai Benjamini and Oded Schramm, Department of Mathematics, The Weizmann Institute, Rehovot 76100, Israel, e-mails: itai@wisdom.weizmann.ac.il, schramm@wisdom.weizmann.ac.ilIL

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