On the spectra of nonsymmetric Laplacian matrices
type: Item
instance of: scholarly article
title: On the spectra of nonsymmetric Laplacian matrices
zbMATH Open document ID: 1076.15012
published in: Linear Algebra and its Applications
publication date: 2005-04-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0508176
review text: A Laplacian matrix \(L = (l_{ij}) \in \mathbb{R}^{n \times n}\) has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a wieghted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with \(-\frac{1}{n} \leq l_{ij} \leq 0\) whenever \(i \neq j\). The authors study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. They also prove that the standardized Laplacian matrices are semiconvergent. Moreover, the multiplicities of the eigenvalues \(0\) and \(1\) 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.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 2160186
zbMATH Keywords: Laplacian matrix
zbMATH Keywords: Laplacian spectrum of a graph
zbMATH Keywords: weighted directed graph
zbMATH Keywords: forest dimension of digraph
zbMATH Keywords: stochastic matrix
author: R. P. Agaev
author: Pavel Chebotarev
MaRDI profile type: Publication
OpenAlex ID: W2093520943
arXiv ID: math/0508176
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DOI: 10.1016/J.LAA.2004.09.003
description: scientific article