On the eigenvalue of the normalized graph Laplacian

Question

Let $0=a_{0}<a_1\leq a_2\leq\cdots\leq a_{N-1}\leq 2$ be a sequence of $N$ real numbers lying in $[0,2]$. Is it true that there exists a connected, finite, simple and undirected graph $G$ with $N$ vertices such that $a_i=\lambda_i$ for $i=0,\ldots, N-1$, where $\lambda_0<\lambda_1\leq \lambda_2\leq\cdots \leq \lambda_{N-1}$ is the spectrum of the normalized graph Laplacian of $G$ ?