David Tweed, I'm glad to hear you comments on this topic.
> Firstly note that both these matrices are hermitian, so all eigenvalues are real and all eigenvectors are orthogonal.
True, for the case of undirected graphs.
> One of the things about the Laplacian matrix is “diagonally dominant”, which amongst other things means the spectrum is well defined.
Doesn't every matrix have a spectrum, which is just defined to be its set of eigenvalues?
Do you have any good introductory references on the subject? So far in the Estrada book, I've found lots of interesting statements about the spectrum of the adjacency matrix, but it doesn't pin down the answer to my initial question. Suppose my graph has two nodes, and the vector (8.8, 14.1) is an eigenvector of the adjacency matrix with eigenvalue 5.9. What is that telling me? You seem to be getting closer to my query when you said:
> Now if you scale the edges of the adjacency graph there’s the straightforward connection that an eigenvector corresponds to the probability of where a particle will end up after a huge number of “uniformly random” steps if it’s initial distribution is uniform over all the nodes. This is of practical use, since if you think that the eigenvector of an enormous adjacency matrix is a solution, you can get an approximate eigenvector by summing lots of a random walks statistics.
Can you spell this out a bit more. What do you mean by scaling the edges of the adjacency graph? What do you mean by taking uniformly random steps; what is the setup for these steps? It's a new terrain for me, so I'm not able to read between the lines too much here.