Editing Discrete Laplace operator (section)

===Graph Laplacians===
There are various definitions of the ''discrete Laplacian'' for [[Graph (discrete mathematics)|graphs]], differing by sign and scale factor (sometimes one averages over the neighboring vertices, other times one just sums; this makes no difference for a [[regular graph]]). The traditional definition of the graph Laplacian, given below, corresponds to the '''negative''' continuous Laplacian on a domain with a free boundary.

Let <math>G = (V,E)</math> be a graph with vertices <math>V</math> and edges <math>E</math>. Let <math>\phi\colon V\to R</math> be a [[function (mathematics)|function]] of the vertices taking values in a [[ring (mathematics)|ring]]. Then, the discrete Laplacian <math>\Delta</math> acting on <math>\phi</math> is defined by

:<math>(\Delta \phi)(v)=\sum_{w:\,d(w,v)=1}\left[\phi(v)-\phi(w)\right]</math>

where <math>d(w,v)</math> is the [[Distance (graph theory)|graph distance]] between vertices w and v. Thus, this sum is over the nearest neighbors of the vertex ''v''. For a graph with a finite number of edges and vertices, this definition is identical to that of the [[Laplacian matrix]]. That is, <math> \phi</math> can be written as a column vector; and so <math>\Delta\phi</math> is the product of the column vector and the Laplacian matrix, while <math>(\Delta \phi)(v)</math> is just the ''v'''th entry of the product vector.

If the graph has weighted edges, that is, a weighting function <math>\gamma\colon E\to R</math> is given, then the definition can be generalized to

:<math>(\Delta_\gamma\phi)(v)=\sum_{w:\,d(w,v)=1}\gamma_{wv}\left[\phi(v)-\phi(w)\right]</math>

where <math>\gamma_{wv}</math> is the weight value on the edge <math>wv\in E</math>.

Closely related to the discrete Laplacian is the '''averaging operator''':

:<math>(M\phi)(v)=\frac{1}{\deg v}\sum_{w:\,d(w,v)=1}\phi(w).</math>