Edit: The views/answers ratio on this question tells me that it was too long. As such, I've stripped out examples and now ask the question in brief. For examples, please ask in the comments or look at the previous revision of this question.
The question:
Consider some operation that maps the set of $n \times n$ symmetric matrices onto into itself $S: \text{sym}_n \rightarrow \text{sym}_n$. Examples include polynomials of a symmetric matrix etc (and, as a result, other functions such as sin and cos).
If we use such an operation on the laplacian $L(G)$ of a graph $G$ (with weighted vertices and edges), the result is a matrix that should also describe a variation of $G$ but with different vertex weights and edges. As such, we can define an operation which maps a graph onto another, by proxy. $\tilde{S}: G \rightarrow \tilde{S}(G), ~ L(\tilde{S}(G)) \equiv S(L(G)).$
For the example of a polynomial $S(L(G)) = \sum_i a_i (L(G))^i$, the spectrum of $L(\tilde{S}(G))$ can be easily found by applying the same polynomial to each eigenvalue of $L(G)$. Polynomials of some simple graphs can in fact be very interesting, and the eigenspectrum of such graphs can be found simply by finding the spectrum of $L(G)$.
Is anybody aware of any books or papers which explore such operations on graphs? I have been looking for some kind of study or reference for this, but to no avail. I have found such graph mappings very useful in some physics research, but I want to make sure it's (a) consistent and (b) not re-inventing the wheel.
