1. Try it! Use the eigenvalue method and compare it for finding sparse cuts in these. You can compare to METIS. Probably, you will get killed (and be slow) compared to METIS. Explain why.
2. Evilness. Construct a family of examples that makes an eigenvalue approach or METIS perform extremely badly (is not within any constant factor of optimal.) Generate them, and show me. Try to understand why.
3. Theory. Read and understand this paper. Perhaps make some progress on the main conjecture.
4. Understand and extend, experiment with ideas in this paper on rounding spectral cuts.