Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any pair of vertices.

The adjacency matrix is \begin{equation} a_{ij} = \begin{cases} 1 & If \, (i,j) \in E \\ 0 & \textrm{Otherwise} \end{cases}, \end{equation} and the Laplacian is $L = \delta_{ij} \sum_{k} a_{ik} - a_{ij}$.

It is known that the number of connected components of $G$ is equal to the multiplicity of the zero eigenvalue of $L$.

Is this result true also for weighted undirected graphs, where \begin{equation} a_{ij} = \begin{cases} w_{ij} & If \, (i,j) \in E \\ 0 & \textrm{Otherwise} \end{cases}, \end{equation} with $w_{ij}$ some positive weight? If so, may you please give me a reference where I can find the proof for weighted graph?

Thank you