Let $G$ be an $n$-vertex graph. Consider the two main matrices associated with the graph: the adjacnecy matrix $A$ and the Laplacian matrix $L=D-A$ (where $D$ is the diagonal matrix of corresponding vertex degrees. As I understand it, the choice of which matrix to use depends upon the precise problem at hand. If we are interested in random walks on the graph, the adjacency matrix often captures this notion better, while if we are interested in connectivity problems, the Laplacian can be used to obtain Cheeger-type analogues relating its spectrum with conductance or expansion. While the two are loosely related, it is often convenient to use the appropriate matrices.
In most situations, we work with regular graphs of degree $d$, say. In this case the adjacency matrix and Laplacian are related as $$L=dI-A$$ which means that for every eigenvalue $\lambda$ of $A$, there is a corresponding eigenvalue $d-\lambda$ of $L$. Moreover, $L$ and $A$ have the same eigenvectors too. So it is easy to shift between the spectra of $L$ and $A$. Everything is cute and simple in the regular case.
But I am interested in the irregular case. Now $A$ and $L$ do not have the same eigenvectors, nor is there any immediate relation between the eigenvalues of $A$ and $L$. I am interested in the possible relations between the spectra of $A$ and $L$.
1) Given $A$, we can certainly compute the spectrum of $L$, and given $L$ we can compute the spectrum of $A$. But given the spectrum of $A$, can we compute the spectrum of $L$? And vice-versa? Is there any explicit correspondence between the spectra of $L$ and $A$ in terms of other possible invariants of the graph?
2) Given $A$ and $L$, what can we say about the spectrum of a linear combination of the form $aA+bL$ for $a,b \in \mathbb{R}$ say?
3) Is there any intuitive interpretation of what the eigenvalues of $A$ and $L$ mean for the graph? As in, do the spectrum of $A$ and the spectrum of $L$ capture different properties of the graph?
After all, $D=diag(A\vec{1})$ so there has to be some useful relations between the spectra of $A$ and $L$ I think/hope. Any pointers would be great!
There are many pairs of graphs that are cospectral relative to the adjacency matrix but not relative to the adjacency matrix. The smallest pair of cospectral graphs on five vertices, for one example. So the answer to (a) is no.
As for (c), there is no known intuitive interpretation of what the eigenvalues of $A$ and $L$ mean for a graph. (Although I think Feynman said that the nice thing about intuition is that it can always be retrained in the light of the examples.)
I am confident that there are graphs that are simultaneously cospectral relative to the adjacency matrix, but not cospectral relative to the unsigned Laplacian. But I'm travelling and cannot undertake the simple search required. So the answer to (b) should be no.