Normalized graph Laplacians for directed graphs

TL;DR

This paper extends the normalized Laplace operator to directed graphs with positive and negative weights, characterizing acyclic graphs and relating eigenvalues to structural properties, with applications to eigenvalue estimation.

Contribution

It introduces a generalized normalized Laplace operator for directed graphs, characterizes acyclic graphs, and establishes eigenvalue comparison theorems linking directed and undirected graphs.

Findings

Characterization of directed acyclic graphs via eigenvalues

Comparison theorems relating directed and undirected graph eigenvalues

Introduction of neighborhood graphs for eigenvalue estimation

Abstract

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.