On the spectra of nonsymmetric Laplacian matrices Original Research Article

A Laplacian matrix, image, 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 image whenever j ≠ i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices image are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of image are equal to the in-forest dimension of the corresponding digraph and one less than the in-forest dimension of the complementary digraph, respectively. We localize the spectra of the standardized Laplacian matrices of order n and study the asymptotic properties of the corresponding domain. One corollary is that the maximum possible imaginary part of an eigenvalue of image converges to image as n → ∞.