Editing Laplacian matrix (section)

=== Laplacian matrix ===
The Laplacian matrix is defined by
: <math>L = D - A, </math>

where ''D'' is the [[degree matrix]] and ''A'' is the [[adjacency matrix]] of the graph. 

For [[directed graph]]s, either the [[degree (graph theory)|indegree or outdegree]] might be used, depending on the application, as in the following example:
{|class="wikitable"
! [[Adjacency matrix]]
! In-Degree matrix
! In-Degree Laplacian
! Out-Degree matrix
! Out-Degree Laplacian
|-
| <math display="inline">\left(\begin{array}{rrr}
  0 &  1 &  2\\
  3 &  0 &  5\\
  6 &  7 &  0\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
  9 &  0 &  0\\
  0 &  8 &  0\\
  0 &  0 &  7\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
  9 &  -1 &  -2\\
  -3 &  8 &  -5\\
  -6 &  -7 &  7\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
  3 &  0  &  0\\
  0 &  8 &  0\\
  0 &  0 &  13\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
  3 &  -1 &  -2\\
  -3 &  8 &  -5\\
  -6 &  -7 &  13\\
\end{array}\right)</math>
|}

Graph self-loops, manifesting themselves by non-zero entries on the main diagonal of the adjacency matrix, are allowed but do not affect the graph Laplacian values.