Eigenvalues of the normalised Laplacian, in increasing order

The spectrum lies in [0, 2] for every graph, and the multiplicity of the eigenvalue 0 is the number of connected components.

Usage

Lapspec(W, tol = 1e-10)

Arguments

W

Symmetric non-negative weight matrix.

tol

An eigenvalue below this counts as zero.

Value

List with values (increasing), lambda1, n_components, max_value, n.

Details

Formula: spectrum of Lcal = T^-1/2 (T - W) T^-1/2, lambda_0 = 0 <= lambda_1 <= ... <= lambda_{n-1} <= 2

References

Chung (1997), Spectral Graph Theory, CBMS 92, Sections 1.2-1.3: the eigenvalues of Lcal lie in [0, 2] and the multiplicity of 0 equals the number of connected components.

Examples

Lapspec(W = 5L)
#> $values
#> [1] 0
#> 
#> $lambda1
#> [1] NaN
#> 
#> $n_components
#> [1] 1
#> 
#> $max_value
#> [1] 0
#> 
#> $n
#> [1] 1
#> 
#> $method
#> [1] "Spectrum of the normalised Laplacian"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"