When you work with weighted graphs you can compute the degree matrix from the weighted adjacency matrix, some time it is good to have weights because they hide geometric information. Moreover, if you have the weighted adj matrix computing degree matrix using the binary form of your weighted adj matrix is easy. In addition, I think your question has more theoretical (e.g. Mathoverflow) than programming foundation (e.g stackoverflow) ;). In any case, you should consult this link for more intuitive explanation of L and its geometric relation.
Good luck :)