I thought "Wow!" when I learned that the eigenvector of the adjacency matrix of a cycle graph $C_n$ corresponding to the second largest eigenvalue gives the coordinates of the vertices when equally distributed on the unit cycle: the $n$-th roots of unity (from complex analysis) come up in a completely discrete context! What's more: The coordinates of the eigenvector can be "interpreted" straight forwardly when assigned properly to the vertices.

There's another straight-forward interpretation of an adjacency eigenvector: the eigenvector corresponding to the largest eigenvalue gives the relative importances of the vertices, being proportional to the sum of the relative importances of its neighbors.

Question: Can more - or eventually all - adjacency eigenvectors be sensibly "interpreted"?

Or does it in general depend on the type of graph, whether and how the eigenvectors can be interpreted, and the first example above is just a curio?

What about the interpretation of the eigen-values? Do at least some of them "mean" something conceivable?