Let me be more specific: If $A=BC$, where $A$ and $C$ are given **Laplacian matrices**, how to calculate $B$? The graph corresponding to $A$ is a **directed ring**, which is strongly connected and $1_n$ and $1_n^T$ are right and left eigenvectors respectively. The graph corresponding to $C$ is a weighted **directed ring**, which is strongly connected but $1_n^T$ is no longer its left eigenvecor while $1_n$ is still its right eigenvector.

For example, $A=\left[ \begin{array}{ccc} 1&-1&0\\0&1&-1\\-1&0&1\end{array} \right]$, that is , $A$ is a circulant ,singular, Laplacian matrix. $C=\left[ \begin{array}{ccc} 1&-1/2&-1/2\\-2/3&1&-1/3\\-4/5&-1/5&1\end{array} \right]$ (singular, **non-symmetric** Laplacian matrix). Then how to compute $B$ if $A=BC$.

Note that $A, B, C$ are all **square matrices**. I **don't** want numerical solutions. There may be many solutions to this problem, so is there a formulated way to find **one** of them (maybe we can restrict $B$ to be Laplacian as well)?