Normalised Laplacian in Chung's sense

Chung's convention T^-1(v, v) = 0 for an isolated vertex is followed exactly, so an isolated vertex contributes a zero row and column.

Usage

Normlap(W)

Arguments

W

Symmetric non-negative weight matrix.

Value

List with Lcal, degree, isolated, n.

Details

Formula: Lcal = T^-1/2 L T^-1/2, i.e. Lcal(u, v) = 1 - w(v, v)/d_v if u = v and d_v != 0, = -w(u, v)/sqrt(d_u d_v) if u ~ v, else 0

References

Chung (1997), Spectral Graph Theory, CBMS 92, Sections 1.2 and 1.4: "L = T^-1/2 L T^-1/2 with the convention T^-1(v, v) = 0 for d_v = 0".

Examples

Normlap(W = 5L)
#> $Lcal
#>      [,1]
#> [1,]    0
#> 
#> $degree
#> [1] 5
#> 
#> $isolated
#> integer(0)
#> 
#> $n
#> [1] 1
#> 
#> $method
#> [1] "Normalised Laplacian T^-1/2 L T^-1/2"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"