ar X iv : m at h / 05 08 17 6 v 1 [ m at h . C O ] 1 0 A ug 2 00 5 On the spectra of nonsymmetric Laplacian matrices ∗
نویسندگان
چکیده
A Laplacian matrix, L = (lij) ∈ R , 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 − 1 n ≤ lij ≤ 0 at j 6= i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices L̃ are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of L̃ 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 L̃ converges to 1 π as n → ∞.
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