Article ID Journal Published Year Pages File Type
9498483 Linear Algebra and its Applications 2005 12 Pages PDF
Abstract
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 → ∞.
Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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