# laplacian_matrix

##### Graph Laplacian

The Laplacian of a graph.

- Keywords
- graphs

##### Usage

```
laplacian_matrix(
graph,
normalized = FALSE,
weights = NULL,
sparse = igraph_opt("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.

##### Examples

```
# NOT RUN {
g <- make_ring(10)
laplacian_matrix(g)
laplacian_matrix(g, norm=TRUE)
laplacian_matrix(g, norm=TRUE, sparse=FALSE)
# }
```

*Documentation reproduced from package igraph, version 1.2.6, License: GPL (>= 2)*