Admittance matrix

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

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 L:=(l_{i,j})_{n \times n} is defined as

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.

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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See also: Admittance matrix, Adjacency matrix, Degree (graph theory), Degree matrix, Discrete Laplace operator, Graph (mathematics), Graph theory, Kirchhoff's theorem, Mathematics