Consider the inverse of the discrete Laplacian $L$ of a graph $G$, $$L^{-1} = (D - A)^{-1}$$ where $D$ is the degree matrix (a matrix whose diagonal entries are the vertex degrees, and the other entries are 0), and $A$ is the adjacency matrix.
Can I write $L^{-1}$ in terms of $D$ and $A$? This would allow one to link the structure of the graph (degrees and adjacency) to the inverse of the Laplacian. The inverse of a sum of two matrices is not usually available in a straightforward way, but perhaps there is a theory when we restrict the matrices to be of this nature?
The $3 \times 3$ case gives $$ L = \left( \begin{array}{ccc} d_1 & -a_{12} & -a_{13} \\ -a_{21} & d_2 & -a_{23} \\ -a_{31} & -a_{32} & d_3 \\ \end{array} \right) $$
$$L^{-1} = \frac{1}{|L|}\left( \begin{array}{ccc} d_2 d_3-a_{23} a_{32} & a_{12} d_3+a_{13} a_{32} & a_{13} d_2+a_{12} a_{23} \\ a_{21} d_3+a_{23} a_{31} & d_1 d_3-a_{13} a_{31} & a_{23} d_1+a_{13} a_{21} \\ a_{31} d_2+a_{21} a_{32} & a_{32} d_1+a_{12} a_{31} & d_1 d_2-a_{12} a_{21} \\ \end{array} \right)$$
Is anything elegant known in general that links $A$ and $D$ to $L^ {-1}$?
