On of the Spectra of Nonsymmetric Laplacian Matrices

A Laplacian matrix, , has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with at . We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of are equal to the in-forest dimension of the corresponding digraph and one less than the in-forest dimension of the complementary digraph, respectively. 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. The imaginary parts of the eigenvalues are bounded from above by ; this maximum converges to as