# Laplacian matrix of a graph with negative weights

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.

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What do you mean by "calculate"? – Felix Goldberg Apr 12 '14 at 8:01
@FelixGoldberg I want to write down the Laplacian matrix for a graph with positive and negative weights in an explicit manner. – Dutta Apr 12 '14 at 8:17

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.