On the spectra of nonsymmetric Laplacian matrices
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.
scientific article