I am trying to construct a graph Laplacian matrix from a set of eigenvalue. I've read several papers about inverse eigenvalue problems but to be honest I didn't understand clearly. Could somebody provide me a general idea about inverse eigenvalue algorithms? Which algorithm is the best for me to choose? Or a concrete example. Below is a detailed description of my question:

Suppose we have a connected undirected graph $G$ with $n$ vertices. $L$ is the Laplacian matrix of $G$, then the eigenvalues of $L$ have the form $$ 0 = \lambda_0 \le \lambda_1 ... \le \lambda_n$$ Now if I slightly change $\lambda_1$ with arbitrary amount to $\lambda_1$* (still in the same form) is it possible to construct another graph Laplacian matrix by this new set of eigenvalues $\{$ $\lambda_0$, $\lambda_1$* $...$ $\lambda_n$ $\}$?

I know that only finite number of sets can be the eigenvalues of a graph laplacian matrix. Does the eigenvalues have any particular form to be the eigenvalues of a Laplacian matrix?

Thanks very much!