|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'501'711
Articles rated: 2609
20 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
On the spectra of nonsymmetric Laplacian matrices | | Rafig Agaev
; Pavel Chebotarev
; | | Date: |
10 Aug 2005 | | Journal: | Linear Algebra and Its Applications. 2005. V. 399. 157--168 DOI: 10.1016/j.laa.2004.09.003 | | Subject: | Combinatorics; Spectral Theory MSC-class: 05C50; 05C05; 05C20; 15A18; 15A51 | math.CO math.SP | | Abstract: | A Laplacian matrix is a square real matrix with 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 the absolute values of the off-diagonal entries not exceeding 1/n, where n is the order of the matrix. 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 a standardized Laplacian matrix are equal to the in-forest dimension of the corresponding digraph and one less than the in-forest dimension of the complementary digraph, respectively. These eigenvalues are semisimple. The spectrum of a standardized Laplacian matrix belongs to the meet of two closed disks, one centered at 1/n, another at 1-1/n, each having radius 1-1/n, 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 1/(2n) cot(pi/2n); this maximum converges to 1/pi as n goes to infinity. Keywords: Laplacian matrix; Laplacian spectrum of graph; Weighted directed graph; Forest dimension of digraph; Stochastic matrix | | Source: | arXiv, math.CO/0508176 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER