Suppose that I have weighted undirected graph $G$, and the corresponding adjacency matrix which is a symmetric matrix $A$.
Suppose that the edge between node $i$ and $j$ has weight $w_{ij}$, then $$ A_{ij} = A_{ji} = w_{ij} $$
The diagonal $A_{ii}$ can consist of arbitrary real numbers.
Because $A$ is symmetric, it can be written as $$ A=\sum \sigma_i v_i v_i^T $$ where $\sigma_i$ is an eigenvalue, $v_i$ is an eigenvector, and eigenvectors are orthogonal.
When we consider a 0-1 adjacency matrix indicating whether or not there is an edge, there are a lot of results from spectral graph theory.
But there doesn't seem to be much results when we consider a weighted adjacency matrix.
I'd like to know if there are interpretations of these eigenvalues and eigenvectors in terms of $G$'s properties.