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

• discrete Laplace operator
• Kirchhoff's theorem