Editing Laplacian matrix (section)

=== Directed multigraphs ===
An analogue of the Laplacian matrix can be defined for directed multigraphs.<ref name="Chaiken1978">{{cite journal
 | title = Matrix Tree Theorems 
 | author1=Chaiken, S. | author2=Kleitman, D. | author-link2=Daniel Kleitman
 | journal = Journal of Combinatorial Theory, Series A 
 | volume = 24
 | number = 3
 | pages = 377–381
 | year = 1978
 | issn = 0097-3165
 | doi=10.1016/0097-3165(78)90067-5
| doi-access = free
 }}</ref> In this case the Laplacian matrix ''L'' is defined as
:<math>L = D - A</math>

where ''D'' is a diagonal matrix with ''D''<sub>''i'',''i''</sub> equal to the outdegree of vertex ''i'' and ''A'' is a matrix with ''A''<sub>''i'',''j''</sub> equal to the number of edges from ''i'' to ''j'' (including loops).