Non self-adjoint laplacians on a directed graph

TL;DR

This paper investigates the spectral properties of a non self-adjoint Laplacian on directed graphs with asymmetric weights, establishing conditions under which the spectrum is discrete and analyzing its implications.

Contribution

It introduces a spectral analysis framework for non self-adjoint Laplacians on directed graphs, including isoperimetric inequalities and spectral gap results.

Findings

Spectral properties depend on Kirchhoff assumptions.

Isoperimetric inequalities relate to the numerical range.

Absence of essential spectrum on heavy end directed graphs.

Abstract

We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We analyse spectral properties of this Laplacian under a Kirchhoff assumption. Moreover we establish isoperimet-ric inequalities in terms of the numerical range to show the absence of the essential spectrum of the Laplacian on heavy end directed graphs.