The Laplacian matrix, , of a weighted digraph is a generalization of the Laplacian (Kirchhoff, admittance) matrix of an ordinary graph. For a weighted digraph whose arc weights belong to [0, 1], the standardized Laplacian matrix ′ is divided by , the number of vertices. We show that the spectrum of the stochastic matrix = ′ + , where is the matrix with all entries 1, and the spectrum of ′ can differ by the multiplicities of 0 and 1 only. Furthermore, we establish the relationship between the eigenspaces of ′ and and demonstrate that ′ and are semiconver-gent. The multiplicity of 1 as the eigenvalue of ′ is shown to be one less than the in-forest dimension of the complementary weighted digraph. The spectrum of belongs to the meet of two closed disks, one centered at 1, another at 1 1, each having radius 1 1, and two closed angles, one bounded with two half-lines drawn from 1, another with two half-lines drawn from 0 through certain points. This is a sharpening of the classical result by Dmitriev and Dynkin on the spectrum of matrices with zero row sums and nonpositive off-diagonal entries
