Editing Laplacian matrix (section)

===Symmetric Laplacian for a directed graph===
Just like for simple graphs, the Laplacian matrix of a directed weighted graph is by definition generally non-symmetric. The symmetry can be enforced by turning the original directed graph into an undirected graph first before constructing the Laplacian. The adjacency matrix of the undirected graph could, e.g., be defined as a sum of the adjacency matrix <math>A</math> of the original directed graph and its [[matrix transpose]] <math>A^T</math> as in the following example:
{|class="wikitable"
! [[Adjacency matrix]]
! Symmetrized adjacency matrix
! Symmetric Laplacian matrix
|-
| <math display="inline">\left(\begin{array}{rrr}
  0 &  1 &  1\\
  0 &  0 &  1\\
  1 &  0 &  0\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
  0 &  1 &  2\\
  1 &  0 &  1\\
  2 &  1 &  0\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
   3 &  -1 &  -2\\
  -1 &   2 &  -1\\
  -2 &  -1 &   3\\
\end{array}\right)</math>
|}
where the zero and one entries of <math>A</math> are treated as numerical, rather than logical as for simple graphs, values, explaining the difference in the results - for simple graphs, the symmetrized graph still needs to be simple with its symmetrized adjacency matrix having only logical, not numerical values, e.g., the logical sum is 1 v 1 = 1, while the numeric sum is 1 + 1 = 2.

Alternatively, the symmetric Laplacian matrix can be calculated from the two Laplacians using the [[degree (graph theory)|indegree and outdegree]], as in the following example:
{|class="wikitable"
! [[Adjacency matrix]]
! Out-Degree matrix
! Out-Degree Laplacian
! In-Degree matrix
! In-Degree Laplacian
|-
| <math display="inline">\left(\begin{array}{rrr}
  0 &  1 &  1\\
  0 &  0 &  1\\
  1 &  0 &  0\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
  2 &  0 &  0\\
  0 &  1 &  0\\
  0 &  0 &  1\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
   2 &  -1 &  -1\\
   0 &   1 &  -1\\
  -1 &   0 &   1\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
  1 &  0 &  0\\
  0 &  1 &  0\\
  0 &  0 &  2\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrr}
   1 &  -1 &  -1\\
   0 &   1 &  -1\\
  -1 &   0 &   2\\
\end{array}\right)</math>
|}
The sum of the out-degree Laplacian transposed and the in-degree Laplacian equals to the symmetric Laplacian matrix.