Matrix analysis
The Laplacian spectrum of a graph
SIAM Journal on Matrix Analysis and Applications
Laplace eigenvalues of graphs—a survey
Discrete Mathematics - Algebraic graph theory; a volume dedicated to Gert Sabidussi
The Laplacian Spectrum of a Graph II
SIAM Journal on Discrete Mathematics
Ring structure digraphs: Spectrum of adjacency matrix and application
Automation and Remote Control
Synchronization in networks of linear agents with output feedbacks
Automation and Remote Control
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The Laplace matrix is the matrix L = (驴ij) 驴 驴 n 脳 n with nonpositive off-diagonal elements and zero row sums. A weighted orgraph corresponds to each Laplace matrix, its properties being closely related to the algebraic properties of the Laplace matrix. The normalized Laplace matrix 719-01 is the Laplace matrix where 驴1/n 驴 驴ij 驴 0 for all i 驴 j. The paper was devoted to the spectrum of the Laplace matrices and to the relationship between the spectra of the Laplace and stochastic matrices. The normalized Laplace matrices were proved to be semiconvergent. It was established that the multiplicity of the eigenvalue 0 of the matrix $$\tilde L$$ is equal to the in-forest dimension of the corresponding orgraph, and the multiplicity of the eigenvalue 1 is one less than the in-forest dimension of the complementary orgraph. The spectra of the matrices $$\tilde L$$ belong to the intersection of two circles of radius 1 驴 1/n centered at the points 1/n and 1 驴 1/n, respectively. Additionally, the domain that comprises them is included in the intersection of two angles (defined in the paper) with vertices 0 and 1 and the band $$|\operatorname{Im} (z)|\; \leqslant \tfrac{1}{{2n}}\cot \tfrac{\pi }{{2n}}$$ (at the limit |Im(z)| n-order Laplace matrices was constructed.