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In the mathematical field of graph theory the admittance matrix or Laplacian matrix is a matrix representation 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
- <math>L := D - A<math>
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 <math>L:=(l_{i,j})_{n \times n}<math> is defined as
- <math>l_{i,j}:=\left\{
\begin{matrix}
\deg(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.
<math>
In the case of directed graphs, either the indegree or the outdegree might be used, depending on the application.
See also
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