R: Graph Laplacian

graph.laplacian {igraph}

R Documentation

Graph Laplacian

Description

The Laplacian of a graph.

Usage

graph.laplacian(graph, normalized=FALSE, weights=NULL,
   sparse=getIgraphOpt("sparsematrices"))

Arguments

`graph`	The input graph.
`normalized`	Whether to calculate the normalized Laplacian. See definitions below.
`weights`	An optional vector giving edge weights for weighted Laplacian matrix. If this is `NULL` and the graph has an edge attribute called `weight`, then it will be used automatically. Set this to `NA` if you want the unweighted Laplacian on a graph that has a `weight` edge attribute.
`sparse`	Logical scalar, whether to return the result as a sparse matrix. The `Matrix` package is required for sparse matrices.

Details

The Laplacian Matrix of a graph is a symmetric matrix having the same number of rows and columns as the number of vertices in the graph and element (i,j) is d[i], the degree of vertex i if if i==j, -1 if i!=j and there is an edge between vertices i and j and 0 otherwise.

A normalized version of the Laplacian Matrix is similar: element (i,j) is 1 if i==j, -1/sqrt(d[i] d[j]) if i!=j and there is an edge between vertices i and j and 0 otherwise.

The weighted version of the Laplacian simply works with the weighted degree instead of the plain degree. I.e. (i,j) is d[i], the weighted degree of vertex i if if i==j, -w if i!=j and there is an edge between vertices i and j with weight w, and 0 otherwise. The weighted degree of a vertex is the sum of the weights of its adjacent edges.

Value

A numeric matrix.

Author(s)

Gabor Csardi csardi.gabor@gmail.com

Examples

g <- graph.ring(10)
graph.laplacian(g)
graph.laplacian(g, norm=TRUE)
graph.laplacian(g, norm=TRUE, sparse=FALSE)

[Package igraph version 0.7.0 Index]