Given a sequence $S$ of real numbers $0=e_0\le e_1 \le \dots \le e_n$, does there exist a simple graph G (possibly with real non-negative weights $\omega_{ij}$ assigned to its edges) whose Laplacian matrix $L$ defined as
\begin{equation} L_{ij}= \begin{cases} \sum_{k,\;k\,\text{is nbr of}\,i}\omega_{ik},& \text{if } i=j\\ -\omega_{ij}, & \text{if } i \text{ is nbr of } j\\ 0 & \text{otherwise} \end{cases} \end{equation}
has spectrum given by $S$ ?
Added later : This problem is somewhat related to the problem of Hearing the shape of a drum except that a) The "drum" in this case is a finite graph with its "sound" being the spectrum of its Laplacian b) I am only interested in the existence of a finite graph for a given spectrum without concerning uniqueness.