Essentially as the title, but I'll give a little bit more background.
I have some finite graph $G$ with $n$ vertices and adjacency matrix $A$. Let $D$ be the $n$ by $n$ matrix with the degree of vertex $i$ at the $i,i$ entry, and 0's everywhere. Finally, let $L = D - A$ be the (unnormalized) graph Laplacian of $G$. Next, fix some collection of eigenvectors and eigenvalues of $L$.
My big-picture question is: Under what conditions are there other graphs which share those eigenvectors/values?
With a little more precision: Approximately how many eigenvectors & eigenvalues can be specified before the answer is no? About how many graphs are there when the answer is yes?
It seems likely that the answer would be a little complicated. I know a few special cases (e.g. 2 eigenvectors/values determine cycles completely; on the other hand, as long as the average degree is fairly large, there are generally very many graphs with the same bottom eigenvector). I certainly appreciate hearing about conditions which aren't tight, as long as they are at least a little broad.
I'm interested in the situation where the eigenvectors DON'T determine the graph, so I would also appreciate any literature pointers to `relaxations' of this idea. For example, one could imagine requiring that the Laplacian contracts the eigenvectors by at least a certain amount (this certainly allows many graphs, but that space is pretty big). In another direction, it seems plausible there is some analogue in the language of graphons.
Thanks for any help!