To define the isoperimetric number of a graph, we first define the density
of a cut in the graph.
Let A \subset V such that |A| \leq n/2. Then, the density of
the cut (A, Ac), is
\frac{ \phi(A) = E(A, Ac)} { |A|}.
The isoperimetric number of a graph is
\phi = min_{A \subset V} \phi(A).
The sparsest cut problem is the problem of finding a set A such that \phi(A) = \phi.
The main theorem that we proved in class today was:
2 * \phi >= \lambda_2 >= (\phi)2 /2d,
where d is the maximum degree of the graph.
Actually, we proved something slightly stronger. If x is a vector perpendicular
to the all-1's vector,
then
xTLx / xTx >= (\phi)2 /2d,
Moreover, there exists a number z such that
xTLx / xTx >= (\phi({ i : x_i >= z))2 /2d.
That is, there exists a cut derived from the eigenvector that witnesses the bound on the isoperimetric number.
Some references for the material coverded in class today are:
@inproceedings{Jerrum-Sinclair/88,
AUTHOR={Jerrum, Mark and Sinclair, Alistair},
TITLE={{Conductance} and the rapid mixing property for {Markov} chains:
{The} approximation of the permanent resolved},
BOOKTITLE={{Proceedings} of the 20th {Ann.} {ACM} {Symposium} on {Theory}
of {Computing} ({Chicago,} {IL,} {May} 2-4, 1988)},
ORGANIZATION={ACM},
PAGES={235--244},
PUBLISHER={ACM Press},
YEAR={1988},
ADDRESS={New York}
}
@article{Sinclair-Jerrum/89,
AUTHOR={Sinclair, Alistair and Jerrum, Mark},
TITLE={{Approximative} counting, uniform generation and rapidly mixing
{Markov} chains},
JOURNAL={Inf.~Comput.},
NUMBER={1},
PAGES={93--133},
PUBLISHER={Academic Press},
VOLUME={82},
YEAR={1989},
MONTH={July},
ADDRESS={New York-San Francisco-London-San Diego}
}
@article{Mohar/89,
AUTHOR={Mohar, Bojan},
TITLE={{Isoperimetric} numbers of graphs},
JOURNAL={J. Comb.~Theory Series B},
PAGES={274--291},
PUBLISHER={Academic Press},
VOLUME={47},
YEAR={1989},
ADDRESS={New York-San Francisco-London-San Diego}
}
Spectral Partitioning Works: Planar Graphs and Finite-Element Meshes, by Spielman and Teng.