On the spectra of nonsymmetric Laplacian matrices

By Rafig P. Agaev, Pavel Yu. Chebotarev. 2005

194 citations recorded by the source corpus.

Abstract A Laplacian matrix, L = ( l ij ) ∈ R n × 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 - 1 n ⩽ l ij ⩽ 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 1 π as n → ∞.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b2e2b7f9f5fb7a91337981d660208d70d3921890

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In Geometry and Topology

Citations in the corpus

Distributed LQR Design for Identical Dynamically Decoupled Systems (2008, background): “By Geršgorin’s theorem [27], the real part of each nonzero eigenvalue of is strictly positive.”
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Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization (2015, background): “The algebraic and geometric multiplicities of this eigenvalue coincide [49], and it is simple if and only the graph G[B] is QSC (has oriented spanning tree).”
On the mathematics of emergence (2007, methodology): “Our use of the word “Laplacian” in this context follows the one in [1] rather than the one in [4].”
Multi‐vehicle coordination for double‐integrator dynamics under fixed undirected/directed interaction in a sampled‐data setting (2010, background): “Let the (nonsymmetric) Laplacian matrix L= [ i j ]∈Rn×n associated with A be defined as [29]”
A tutorial on modeling and analysis of dynamic social networks. Part I (2017, background): “An example of a singular M-matrix is the Laplacian (or Kirchhoff ) matrix of a weighted graph [62, 63, 73].”
Sampled-data discrete-time coordination algorithms for double-integrator dynamics under dynamic directed interaction (2010, background): “:3) n associated with A be defined as Agaev and Chebotarev (2005) Zero is an eigenvalue of L with an associated eigenvector 1 n , where 1 n is the n (cid”
Consensus Tracking Under Directed Interaction Topologies: Algorithms and Experiments (2008, background): “Let the (nonsymmetric) Laplacian matrix be defined as [18] and , .”
Coordination in multiagent systems and Laplacian spectra of digraphs (2009, background): “The problem of localizing the spectra of Laplacian matrices was partially solved in [10].”