Spectral Methods have had huge influence of modern algorithm design. For algorithmic problems on graphs, this is done by using a deep connection between random walks and the powers of various natural matrices associated with the graph. The major contribution

of this thesis initiates attempts to recover algorithmic results in Graph Minor Theory via spectral methods.

We make progress towards this goal by exploring these questions in the Property Testing Model for bounded degree graphs. Our main contributions are

1. The first result gives an almost query optimal one-sided tester for the property of H-minor-freeness. Benjamini-Schramm-Shapira (STOC 2008) conjectured that for fixed H, this can be done in time O(sqrt n). Our algorithm solves this in time n^{1/2+o(1)} which nearly resolves this upto n^{o(1)} factors.

2. BSS also conjectured that in the two-sided model, H-minor-freeness can be tested in time poly(1/eps). We resolve this conjecture in the affirmative.

3.Lastly, in a previous work on the two-sided-question above, Hassidim-Kelner-Nguyen-Onak (FOCS 2009) introduced a tool they call partition oracle. They conjectured that partition oracles could be implemented in time poly(1/eps) and gave an implementation which took exp(poly(1/eps)) time. In this work, we resolve this conjecture and produce such an oracle.

Additionally, this work also presents an algorithm which can recover a planted 3-coloring in a graph with some random like properties and suggests some future research directions alongside.