Expanders, Property Testing and the PCP theorem

Lecture 5: Expanders, Probabilistic Proof of Existence

Posted in lectures by HQN on September 14, 2008

We gave a proof showing that most regular graphs are good (edge/vertex) expanders. In terms of edge-expansion, Bella Bollobas in this paper showed that, for every d\geq 3, most random d-regular graphs have edge-expansion rate (i.e. edge-isoperimetric number) at least d/2 - c_1\sqrt d for some constant c_1. (Specifically, we can set c_1=\sqrt{\lg 2}. On the other hand, given any d-regular graph G of order n, if we select a random subset U of vertices of size \lfloor n/2 \rfloor, then the expected number of edges crossing the (U,\bar U) cut is \frac{\frac{dn}{2}\lfloor n/2 \rfloor \lceil n/2 \rceil}{\binom n 2}

So the edge-isoperimetric number is bounded above by \frac d 2 \frac{n+1}{n-1}. Thus, for large graphs the edge expansion rate is (roughly) at most \frac d 2.

Better yet, Noga Alon proved in this paper that there is a constant c_2 such that the edge-isoperimetric number of any d-regular graph (with sufficiently large number of vertices) is at most d/2 - c_2 \sqrt d.

In summary, we can roughly think of edge-isoperimetric numbers of random d-regular graphs to be about d/2 - \Theta(\sqrt d).

For small values of d, Bollobas’ analysis yields the following. For every sufficiently large n,

  • there exists a (n,3,2/11)-edge expander
  • there exists a (n,4,11/25)-edge expander
  • there exists a (n,5,18/25)-edge expander
  • there exists a (n,6,26/25)-edge expander
  • there exists a (n,9,2.06)-edge expander

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