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 |
---|---|
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 |