Normalized graph Laplacians for directed graphs

Details Normalized graph Laplacians for directed graphs Normalized graph Laplacians for directed graphs

2011-07-25

arxiv.org/abs/1107.4847v2

Mathematics

Combinatorics

40 pages, 3 figures

Scientific paper

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.

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