TITLE: Motivation for spectral graph theory.
QUESTION [80 upvotes]: Why do we care about eigenvalues of graphs?
Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so they must be important.
I always assumed that spectral graph theory extends graph theory by providing tools to prove things we couldn't otherwise, somewhat like how representation theory extends finite group theory.  But most results I see in spectral graph theory seem to concern eigenvalues  not as means to an end, but as objects of interest in their own right.
I also considered practical value as motivation, e.g. using a given set of eigenvalues to put bounds on essential properties of graphs, such as maximum vertex degree.  But I can't imagine a situation in which I would have access to a graph's eigenvalues before I would know much more elementary information like maximum vertex degree.
(EDIT: for example, dtldarek points out that $\lambda_2$ is related to diameter, but then why would we need $\lambda_2$ when we already have diameter?  Is this somehow conceptually beneficial?)

So, what is the meaning of graph spectra intuitively?  And for what practical purposes are they used?  Why is finding the eigenvalues of a graph's adjacency/Laplacian matrices more than just a novel problem?

REPLY [8 votes]: Another reason the eigenvalues are used if for approximating certain functions in a graph that would be too complex to compute otherwise.  For instance, in social networks one often wants to count the number of paths of length $k$ between two nodes $u$ and $v$.  In a large graph, this may be too complex to compute exactly.  One way to approximate it consists in considering the $k^{\mathrm{th}}$ power of the adjacency matrix $\mathbf A$. However, even if $\mathbf A$ is sparse, $\mathbf A^k$ is not for even small $k$ (remember the "six degrees of separation" – the diameter of a social network is low). Thus, a method to compute entries of $\mathbf A^k$ approximately uses the eigenvalue decomposition of $\mathbf A$.  If the graph is undirected then we can write
$$
\mathbf A = \mathbf U \mathbf Λ \mathbf U^{\mathrm T}
$$
with $\mathbf U$ and $\mathbf Λ$ being $n \times n$ matrices.
A rank-reduced eigenvalue can be computed efficiently, i.e., one where $\mathbf U$ is replaced by an $n×k$ matrix $\mathbf {\tilde U}$, and $\mathbf Λ$ by a  $k×k$ matrix $\mathbf{\tilde Λ}$.  Then, $\mathbf A^k$ can be approximated by 
$$
\mathbf A^k ≈ \mathbf{\tilde U} \mathbf{\tilde Λ}^k \mathbf{\tilde U}^{\mathrm T}
$$
and to approximate only $(\mathbf A^k)_{uv}$ (i.e., the number of $k$-paths between nodes $u$ and $v$), we only need to compute $\mathbf{\tilde U}_{i\bullet} \mathbf{\tilde D}^k \mathbf{\tilde U}_{j\bullet}^{\mathrm T}$.  This is a case where one is not interested in the eigenvalues of eigenvectors themselves, but uses the eigenvalue decomposition for approximation.
The same technique can be used to approximate other functions of $\mathbf A$, for instance the matrix exponential.