Symmetric normalised Laplacian matrix

Question

I have the relation for symmetric Laplacian matrix: \begin{equation} L=I-D^{-1/2}AD^{-1/2} \end{equation}

I know the matrix $L$ and now I am wondering if I can use the above relationship to calculate matrix $A$. Obviously $D$ is unknown as well.

Thanks in advanced

Answer 1

Let us consider the unweighted graph without loops. As you know L, you could infer $D$ by merely calculating the non-zero non-diagonal elements for each row $L_{ij} = - \frac{1}{\sqrt{d_i d_j}}$. I.e., $d_1$ will be the number of non-zero $L_{1j}, j\neq1$.

The further is simple: $$ A = D - \sqrt{D}L \sqrt{D}$$.