The Laplacian of a graph.

## Usage

```
laplacian_matrix(
graph,
weights = NULL,
sparse = igraph_opt("sparsematrices"),
normalization = c("unnormalized", "symmetric", "left", "right"),
normalized
)
```

## Arguments

- graph
The input graph.

- 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.- normalization
The normalization method to use when calculating the Laplacian matrix. See the "Normalization methods" section on this page.

- normalized
Deprecated, use

`normalization`

instead.

## 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.

The Laplacian matrix can also be normalized, with several conventional normalization methods. See the "Normalization methods" section on this page.

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.

## Normalization methods

The Laplacian matrix \(L\) is defined in terms of the adjacency matrix \(A\) and a diagonal matrix \(D\) containing the degrees as follows:

"unnormalized": Unnormalized Laplacian, \(L = D - A\).

"symmetric": Symmetrically normalized Laplacian, \(L = I - D^{-\frac{1}{2}} A D^{-\frac{1}{2}}\).

"left": Left-stochastic normalized Laplacian, \({L = I - D^{-1} A}\).

"rigth": Right-stochastic normalized Laplacian, \(L = I - A D^{-1}\).

## Author

Gabor Csardi csardi.gabor@gmail.com

## Examples

```
g <- make_ring(10)
laplacian_matrix(g)
#> 10 x 10 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 2 -1 . . . . . . . -1
#> [2,] -1 2 -1 . . . . . . .
#> [3,] . -1 2 -1 . . . . . .
#> [4,] . . -1 2 -1 . . . . .
#> [5,] . . . -1 2 -1 . . . .
#> [6,] . . . . -1 2 -1 . . .
#> [7,] . . . . . -1 2 -1 . .
#> [8,] . . . . . . -1 2 -1 .
#> [9,] . . . . . . . -1 2 -1
#> [10,] -1 . . . . . . . -1 2
laplacian_matrix(g, normalization = "unnormalized")
#> 10 x 10 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 2 -1 . . . . . . . -1
#> [2,] -1 2 -1 . . . . . . .
#> [3,] . -1 2 -1 . . . . . .
#> [4,] . . -1 2 -1 . . . . .
#> [5,] . . . -1 2 -1 . . . .
#> [6,] . . . . -1 2 -1 . . .
#> [7,] . . . . . -1 2 -1 . .
#> [8,] . . . . . . -1 2 -1 .
#> [9,] . . . . . . . -1 2 -1
#> [10,] -1 . . . . . . . -1 2
laplacian_matrix(g, normalization = "unnormalized", sparse = FALSE)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 2 -1 0 0 0 0 0 0 0 -1
#> [2,] -1 2 -1 0 0 0 0 0 0 0
#> [3,] 0 -1 2 -1 0 0 0 0 0 0
#> [4,] 0 0 -1 2 -1 0 0 0 0 0
#> [5,] 0 0 0 -1 2 -1 0 0 0 0
#> [6,] 0 0 0 0 -1 2 -1 0 0 0
#> [7,] 0 0 0 0 0 -1 2 -1 0 0
#> [8,] 0 0 0 0 0 0 -1 2 -1 0
#> [9,] 0 0 0 0 0 0 0 -1 2 -1
#> [10,] -1 0 0 0 0 0 0 0 -1 2
```