I would reccomend looking at 'Proto Value Functions' ( http://www.machinelearning.org/.../070_ProtoValue_Mahadevan.... )[1]; they use eigenvectors of adjaceny matrices for reinforcement learning --- but because gridworld has a simple adjacency martix, the resulting eigenvectors are somewhat human-readable, and the pictures help a lot. If you're really curious, I can probably bang up some python or MATLAB or something.

The essential intuition is that higher numbered eigenvectors represent more local adjacency information --- so the first eigenvector deals with global connectivity (the relative value of i-th and j-th component says something about how interconnected nodes i and j are in terms of the whole graph). A higher eigenvalue's corresponding vector has a higher local gradient, and the direction of the gradient is less constant across the graph (there is not necessarily a monotonic path from node to node) so says less about global connectivity --- but has a higher granularity when examining local connectivity.

1[Edited to warn that Mahadevan's papers are fantastically dense --- and you don't need to understand the paper to gain intuition from the pretty pictures; read the text at your own risk]