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

    \begin{align*} 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} \end{align*}

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