The Laplacian of a graph.
laplacian_matrix(graph, normalized = FALSE, weights = NULL, sparse = igraph_opt("sparsematrices"))
- The input graph.
- Whether to calculate the normalized Laplacian. See definitions below.
- An optional vector giving edge weights for weighted Laplacian
matrix. If this is
NULLand the graph has an edge attribute called
weight, then it will be used automatically. Set this to
NAif you want the unweighted L
- Logical scalar, whether to return the result as a sparse
Matrixpackage is required for sparse matrices.
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.
- A numeric matrix.
g <- make_ring(10) laplacian_matrix(g) laplacian_matrix(g, norm=TRUE) laplacian_matrix(g, norm=TRUE, sparse=FALSE)