I have a $(q^2+q)(q+1)$-regular graph. Is there some general method to compute the eigenvalues of the adjacency matrix of a $k$-regular graph explicitly? Or could we estimate its eigenvalues? Thank you very much.
Edit: I would like to know the bound of $$max_i(|\lambda_i|),$$ where $\lambda_i$ runs over all eigenvalues except the ones with $|\lambda|=k$.
Your graph is indeed nice. In fact it's a Grassmann graph, also known as a Grassmann (association) scheme or $q$-Johnson graph. It's especially nice in your case because it's strongly-regular and so the eigenvalues can be written down explicitly. Fortunately it's already done here. In particular, the eigenvalues of your graph are $(q^2+q)(q+1)$, $q^2-1$ and $-q-1$.