Suggestion for framing a course in Representation theory + Spectral graph theory I am going to give a course in spectral graph theory to graduate students. I want to learn and teach the connection between the spectral graph theory and the representation theory of finite groups. I am good at both the areas but I am not sure where to start and what to include. It would be a great help to me if you can suggest what to add beyond the basics in both the areas. I haven't seen any book on the connection between these areas.
Some things in my mind are:

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*Representation theory of $Aut(G)$ where $G$ is a finite simple graph.


*Reducible/Irreducible eigenspaces.
I am not aware of the literature where representation theory is used in spectral graph theory or vice versa. Kindly give your suggestions.
Thank you.
 A: 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:

*

*[1] directly related: Representations of the automorphism group of graphs via spectral graphs theory

*[2] how to get the irreducible subspaces when the eigenspaces are not irreducible: Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix

*[3] a simple construction (the arrangement space) that I always found helpful for organizing my thoughts when working in spectral graph theory, representation theory and geometry at the same time (check in particular the two last bullet points): Where have you encountered "arrangement spaces"?
A: A paper that might be suitable for your course, that touches on both the points you listed, is Graph Automorphisms from the Geometric Viewpoint. As the abstract says, it's concerned with the representation theory of $Aut(G)$ where $G$ is a finite graph.
Another collection of literature that might go along with the course you sketch is the theory of Quiver representations.
Lastly, Daniel Spielman wrote a book on Spectral and Algebraic Graph Theory. While not tied directly to representations of $Aut(G)$, this might have some ideas that could help as you plan your course. It sounds like a great course. Good luck!
