Laplacian matrix of a graph
Let be a finite graph with vectices and let be the incidence matrix (http://planetmath.org/ IncidenceMatrixWithRespectToAnOrientation) of with respect to some orientation. The Laplacian matrix of is defined to be .
If we let be the adjacency matrix of then it can be shown that , where and is the degree of the vertex . As a result, the Laplacian matrix is independent of what orientation is chosen for .
The Laplacian matrix is usually denoted by . It is a positive semidefinite singular matrix, so that the smallest eigenvalue is 0.
Title | Laplacian matrix of a graph |
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Canonical name | LaplacianMatrixOfAGraph |
Date of creation | 2013-03-22 17:04:40 |
Last modified on | 2013-03-22 17:04:40 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 8 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 05C50 |
Defines | Laplacian matrix |