Ok, I think I see your point: if you scale the rows of the adjacency matrix so that it becomes a stochastic matrix, then the fixed points of matrix have the meaning as attractor points for the corresponding Markov chain. So this is an example where eigenvectors with eigenvalue 1 have a clear meaning. But it is for a different matrix than the adjacency matrix (which is not generally a scalar multiple of it), and it concerns the specific eigenvalue 1 -- so my question is still open about the meaning of general eigenvectors of the adjacency matrix.