Is there a good intuition behind why the eigenvalues of a matrix corresponding to a graph tell us useful information about the graph? There are a lot of results relating the eigenvalues of the adjacency matrix of a graph to the graph, and similar statements about the eigenvalues of the Laplacian of the graph or the normalized Laplacian.
The best intuition I have is something like this: Given a graph $G$ with adjacency matrix $M$, one can imagine unicelled organisms on each vertex. At each stage, each organism reproduces and sends a copy of itself to each adjacent vertex. Then, given a given list $\vec{v}$ of initial numbers of organisms on each vertex, $M^k \vec{v}$ tells us how many are on each vertex after $k$ steps. A suitably normalized version of this then corresponds to the ratios of the populations at each vertex, and that connects with the eigenvalues and eigenvectors.
One can tell a somewhat similar story of random walks and the Laplacian.
But these seem very handwavy. Is there a better explanation for why the eigenvalues of these matrices are so useful?
