Normalized graph Laplacians for directed graphs
Copyright policy )Normalized graph Laplacians for directed graphs
We consider the normalized Laplace operator for directed graphs with positive and negative edge weights. This generalization of the normalized Laplace operator for undirected graphs is used to characterize directed acyclic graphs. Moreover, we identify certain structural properties of the underlying graph with extremal eigenvalues of the normalized Laplace operator. We prove comparison theorems that establish a relationship between the eigenvalues of directed graphs and certain undirected graphs. This relationship is used to derive eigenvalue estimates for directed graphs. Finally we introduce the concept of neighborhood graphs for directed graphs and use it to obtain further eigenvalue estimates.
40 pages, 3 figures
Numerical Analysis, Algebra and Number Theory, Graphs and linear algebra (matrices, eigenvalues, etc.), directed acyclic graphs, eigenvalues, Directed graphs (digraphs), tournaments, Eigenvalues, directed graphs, Directed acyclic graphs, Mathematics - Spectral Theory, normalized graph Laplace operator, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Geometry and Topology, Combinatorics (math.CO), Normalized graph Laplace operator, Neighborhood graph, Spectral Theory (math.SP), Directed graphs, neighborhood graph
Numerical Analysis, Algebra and Number Theory, Graphs and linear algebra (matrices, eigenvalues, etc.), directed acyclic graphs, eigenvalues, Directed graphs (digraphs), tournaments, Eigenvalues, directed graphs, Directed acyclic graphs, Mathematics - Spectral Theory, normalized graph Laplace operator, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Geometry and Topology, Combinatorics (math.CO), Normalized graph Laplace operator, Neighborhood graph, Spectral Theory (math.SP), Directed graphs, neighborhood graph
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