On the spectra of nonsymmetric Laplacian matrices

A Laplacian matrix, L=(ℓij)∈Rn×n, 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 -1n⩽ℓij⩽0 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 L∼ are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of L∼ 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 L∼ converges to 1π as n → ∞.