I am trying to calculate the Laplacian and Adjacency matrix of a graphs for positive and negative weights. If a graph be simple with only non-negative weight it is easier. But in my graph I have some negative weights and loops. 


Please give me some references and hints if available.


Thank you for your help. 



Laplacian matrix of a graph with negative weights
The most natural definition of Laplacian matrix is to me $\mathcal L=\mathcal I\mathcal I^T$, where $\mathcal I$ is the incidence matrix of an arbitrary orientation of the graph; or more generally $\mathcal L=\mathcal I\mathcal M\mathcal I^T$, where $\mathcal M$ is the diagonal matrix whose entries are the edge weights. Now, the very same definition can be used to define a Laplacian matrix with general (i.e., also negative) weights.


