Laplacian matrix of a graph

Another important symmetric matrix associated with a graph is the Laplacian matrix. This is the matrix L = A^TA, with A as the arc-node incidence matrix. It can be shown that the (i, j) element of the Laplacian matrix is given by

L_{i j}= \begin{cases}\# \operatorname{arcs} \text { incident to node } i & \text { if } i=j, \\ -1 & \text { if there is an arc joining node } i \text { to node } j, \\ 0 & \text { otherwise. }\end{cases}

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Hyper-Textbook: Optimization Models and Applications Copyright © by L. El Ghaoui. All Rights Reserved.

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