Laplacian matrix

In the mathematical field of graph theory the admittance matrix or laplacian matrix is a matrix represenatition of a graph. Together with Kirchhoff's theorem it can be used to calculate the number of spanning trees for a given graph.

Definition

The admittance matrix of a graph G is defined as

$L:=D-A$

with D the degree matrix of G and A the adjacency matrix of G.

More explicitly, given a graph G with n vertices the admittance matrix of G is

$l_{i,j}:=\left\{{\begin{matrix}{\mbox{degree}}(v_{i})&{\mbox{if}}\ i=j\\-1&{\mbox{if}}\ i\neq j\ {\mbox{and}}\ v_{i}\ {\mbox{adjacent}}\ v_{j}\\0&{\mbox{otherwise}}\end{matrix}}\right.$

See also

• discrete Laplace operator
• Kirchhoff's theorem