Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9498483 | Linear Algebra and its Applications | 2005 | 12 Pages |
Abstract
A Laplacian matrix, L=(âij)âRnÃn, 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⩽âij⩽0 whenever j â  i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices Lâ¼ are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of Lâ¼ 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 Lâ¼ converges to 1Ï as n â â.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Rafig Agaev, Pavel Chebotarev,