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 matrixComment: 11 page

We study the matrices Q_k of in-forests of a weighted digraph G and their connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of L. The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the Laplacian matrix, which provides an immediate proof of the Markov chain tree theorem. A source of these results is the fact that matrices Q_k are the matrix coefficients in the polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's matrices for -L. Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection; Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic Graph Theor

Abstract-Constructing and studying distributed control systems requires the analysis of the Laplacian spectra and the forest structure of directed graphs. In this paper, we present some basic results of this analysis partially obtained by the present authors. We also discuss the application of these results published earlier to decentralized control and touch upon some problems of spectral graph theory.

Abstract-In the coordination/consensus problem for multi-agent systems, a well-known condition of achieving consensus is the presence of a spanning arborescence in the communication digraph. The paper deals with the discrete consensus problem in the case where this condition is not satisfied. A characterization of the subspace T P of initial opinions (where P is the influence matrix) that ensure consensus in the DeGroot model is given. We propose a method of coordination that consists of: (1) the transformation of the vector of initial opinions into a vector belonging to T P by orthogonal projection and (2) subsequent iterations of the transformation P. The properties of this method are studied. It is shown that for any non-periodic stochastic matrix P, the resulting matrix of the orthogonal projection method can be treated as a regularized power limit of P.

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