Meaning of Eigenvectors in a Graph

[gigaJoule]

What interpretation could the eigenvectors in a graph have? By graph I mean an adjaceny matrix not counting self-loops. If you can draw any physical meanings or point to any examples, that'd be even better.

Thank you!

 

[Ackbach]

This may not be quite what you're after, but I do know this: the i,j component of

$A^{n}$

represents the number of walks of length n from vertex i to vertex j. Now, if you wish to compute this explicitly, the eigenvalues and eigenvectors will let you do that via diagonalization. So here I'm demonstrating a use for the eigenvectors, but not, perhaps, a physical meaning directly.

The relevant field here is spectral graph theory. Maybe you could find some books in a university library related to spectral graph theory.