The question is a bit vague, but any ideas/directions will be appreciated.

Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ capture "global" information about the graph, akin to a fourier transform. For instance, the number of cycles in $G$ of length $k$ is $Tr(A^k)$ which is the $k$-th power sum of the eigenvalues.

But even this uses all the eigenvalues. In particular, take some eigenvalue $\lambda$ and its eigenvector $\vec{f}$. Does $\lambda^k$ alone have any natural correspondence with some measure on the cycles of length $k$?

Secondly, since the spectrum captures global properties of the graph, like expansion, it seems as if slightly changing the graph locally can change the spectrum and eigenvectors significantly in a global way. This can be checked empirically, though I am unclear on how to formalize this.

Does the reverse hold in any meaningful way? That is, suppose I change the graph "globally", for instance, I change every vertex/edge the same way. Would that change the spectrum and eigenvalues "locally"?

The example that illustrates my vague question is the result of Lubetzky-Peres, which says that going from the adjacency matrix $A$ to the Hashimoto matrix (of the line-digraph of $G$) changes the eigenvalues and eigenvectors locally. Each eigenvalue $\lambda$ of $A$ splits into two eigenvalues $\mu,\overline{\mu}$ which are roots of $$x^2-\lambda x+ (d-1)=0$$ and the eigenvector corresponding to $\mu$ at a "vertex" $(v,w)$ is $\mu f(w)-f(v)$. So modifying the graph "globally" as in the case of the line-digraph construction (where we add two vertices for each direction of each edge of $G$) changes the eigenvalues and eigenvectors only locally.

Is there an underlying principle here that governs the degree of locality/globality of the spectrum and eigenvectors? What are the precise class of operations on a graph that change the spectrum locally?