  Let \phi(G) be the minimum conductance of an undirected graph G, and let
0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the
normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2,
  \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee
is achieved by the spectral partitioning algorithm. This improves Cheeger's
inequality, and the bound is optimal up to a constant factor for any k. Our
result shows that the spectral partitioning algorithm is a constant factor
approximation algorithm for finding a sparse cut if \lambda_k$ is a constant
for some constant k. This provides some theoretical justification to its
empirical performance in image segmentation and clustering problems. We extend
the analysis to other graph partitioning problems, including multi-way
partition, balanced separator, and maximum cut.
