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 the arc-node incidence matrix. It can be shown that the (i,j) element of the Laplacian matrix is given by

 L_{ij} = left{ begin{array}{ll}  mbox{# arcs incident to node } i & mbox{if } i=j , -1 & mbox{if there is an arc joining node } i mbox{ to node } j,  0 & mbox{otherwise.} end{array} right.

See also: