Editing Laplacian matrix (section)

=== Laplacian matrix for an undirected graph via the oriented incidence matrix===
The <math display="inline">|v| \times |e|</math> oriented [[incidence matrix]] ''B'' with element ''B''<sub>''ve''</sub> for the vertex ''v''  and the edge ''e'' (connecting vertices <math display="inline">v_i</math> and <math display="inline">v_j</math>, with ''i''&nbsp;≠&nbsp;''j'') is defined by
:<math>B_{ve} = \left\{\begin{array}{rl}
   1, & \text{if } v = v_i\\
  -1, & \text{if } v = v_j\\
   0, & \text{otherwise}.
\end{array}\right.</math>
Even though the edges in this definition are technically directed, their directions can be arbitrary, still resulting in the same symmetric Laplacian <math display="inline">|v| \times |v|</math> matrix ''L'' defined as
:<math>L = B B^\textsf{T}</math>
where <math display="inline">B^\textsf{T}</math> is the [[transpose|matrix transpose]] of ''B''.

{|class="wikitable"
! [[Undirected graph]]
! [[Incidence matrix]]
! Laplacian matrix
|-
| [[image:Labeled_undirected_graph.svg|100px]]
| <math display="inline">\left(\begin{array}{rrrr}
   1 &  1 &  1 &  0\\
  -1 &  0 &  0 &  0\\
   0 & -1 &  0 &  1\\
   0 &  0 & -1 & -1\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrrr}
   3 &  -1 &  -1 & -1\\
  -1 &   1 &   0 &  0\\
  -1 &   0 &   2 & -1\\
  -1 &   0 &   -1 & 2\\
\end{array}\right)</math>
|}

An alternative product <math>B^\textsf{T}B</math> defines the so-called <math display="inline">|e| \times |e|</math> ''edge-based Laplacian,'' as opposed to the original commonly used ''vertex-based Laplacian'' matrix ''L''.