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Encyclopedia > Laplacian 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. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics and computer science, graph theory studies the properties of graphs. ... For the square matrix section, see square matrix. ... This article just presents the basic definitions. ... A spanning tree (red) of a graph (black), superimposed In the mathematical field of graph theory, a spanning tree of a connected, undirected graph is a tree which includes every vertex of that graph. ...

## Definition

The admittance matrix of a graph G is defined as

L: = DA

with D the degree matrix of G and A the adjacency matrix of G. In mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry aij is the number of edges from vi to vj in . ...

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

In the case of directed graphs, either the indegree or the outdegree might be used, depending on the application. Results from FactBites:

 PlanetMath: Laplacian matrix of a graph (92 words) It is a positive semidefinite singular matrix, so that the smallest eigenvalue is 0. "Laplacian matrix of a graph" is owned by Mathprof. This is version 5 of Laplacian matrix of a graph, born on 2007-05-12, modified 2007-05-14.
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