Update
I have since uploaded a preprint discussing this connection. This is probably not it‘s final form, but since I claimed writing on this some years ago, it is more than time to finally mention it here. Allow me to also point to my PhD thesis (follow the links in my profile) where I explore some of this in more detail.
Otherwise, the answer is unchanged and contains below some of the sources and main ideas I have used in the past.
I alway put my focus on the idea of the graph realization, because it gives the subject a geometric touch. A graph realization is simply a map assigning to each vertex $i\in V$ a point $v_i$ in Euclidean space.
And such a realization can be highly symmetric (related to representation theory) or it can be some sort of balanced configuation (related to spectral graph theory). These ideas are not independent.
For example, suppose you have a realization that satisfies some kind of self-stress condition:
$$(*)\qquad \sum_{j\in N(i)} v_j = \theta v_i\quad\text{for all $i\in V$}.$$
Let $M$ be the matrix in which the $v_i$ are the rows, then you can write $(*)$ as $AM=\theta M$ (where $A$ is the adjacency matrix of the graph). Immediately you see that $\theta$ must be an eigenvalue of $A$, and the columns of $M$ must be eigenvectors.
The columns need not span the whole eigenspace.
But if they do, then we call it a spectral realization (see also the link [1] below).
If you define the arrangement space $U:=\mathrm{span}(M)$ as the column span of $M$ (see also the link [3] below), then you have a handy way to define symmetric and spectral realizations:
- a realization is symmetric if its arrangement space is $\mathrm{Aut}(G)$-invariant.
- a realization is spectral if its arrangement space is an eigenspace of $A$.
And since eigenspaces are always invariant, we immediately find that spectral realizations are always as symmetric as the underlying graph.
In my opinion, it is this property of spectral realizations that tells us a lot about the structure of the graph (at least for highly symmetric graphs).
Others might use them on less symmetric graphs in graph drawing algorithms or optimization (but I feel this is less related to representation theory).
If you take the convex hull of the vertices in a spectral graph realization, you obtain the eigenpolytope of a graph.
The literature on these is quite scattered, but the initial source is probably "Graphs, groups and polytopes" by Godsil (I have since tried to organize the literature in this other (work in progress) preprint).
Godsil proved that the eigenpolytope is as symmetric as the initial graph. He also proves group theoretic properties of $\mathrm{Aut}(G)$ from these polytopes (which are just graph realizations in disguise).
You asked specifically about reducible/irreducible eigenspaces. In general, it is quite tricky to determine whether the eigenspaces of a graph are irreducible (without computing all irreducible subspaces). But there is one case for which it is easy: distance-transitive graphs. For these, the eigenspaces are exactly the irreducible subspaces of $\mathrm{Aut}(G)$. This basically follows from Proposition 4.1.11 (p. 137) in "Distance Regular Graphs" by Brouwer, Cohen and Neumaier.
Their proof is in a purely represenation theoretic language, but in the preprints I also discuss some more elementary approaches.
Finally, I can think about some connections to rigidity theory.
One might consider only the deformations of a graph realization that preserves the symmetry of the structure.
Whether such deformations exist depends on the decomposition of the permutation-representation of $\mathrm{Aut}(G)$ into irreducible representations (in particular, their multiplicities).
To connect this to spectral graph theory, one can observe that if a realization is rigid (i.e. it cannot be deformed without loosing symmetry), and irreducible, then one can show that it satisfies $(*)$ (it is not necessarily spectral, but almost).
Of course, for distance-transitive graphs, this implies that the realization is spectral.
Here are some older posts of mine that might be related:
