Editing Laplacian matrix (section)

== Interpretation as the discrete Laplace operator approximating the continuous Laplacian==
The graph Laplacian matrix can be further viewed as a matrix form of the negative [[discrete Laplace operator]] on a graph approximating the negative continuous [[Laplacian]] operator obtained by the [[finite difference method]].
(See [[Discrete Poisson equation]])<ref>{{citation
 | last1 = Smola | first1 = Alexander J.
 | last2 = Kondor | first2 = Risi
 | contribution = Kernels and regularization on graphs
 | doi = 10.1007/978-3-540-45167-9_12
 | pages = 144–158
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Learning Theory and Kernel Machines: 16th Annual Conference on Learning Theory and 7th Kernel Workshop, COLT/Kernel 2003, Washington, DC, USA, August 24–27, 2003, Proceedings
 | volume = 2777
 | year = 2003
| isbn = 978-3-540-40720-1
 | citeseerx = 10.1.1.3.7020
 }}.</ref> In this interpretation, every graph vertex is treated as a grid point; the local connectivity of the vertex determines the finite difference approximation [[stencil (numerical analysis)|stencil]] at this grid point, the grid size is always one for every edge, and there are no constraints on any grid points, which corresponds to the case of the homogeneous [[Neumann boundary condition]], i.e., free boundary. Such an interpretation allows one, e.g., generalizing the Laplacian matrix to the case of graphs with an infinite number of vertices and edges, leading to a Laplacian matrix of an infinite size.