Non Self-Adjoint Laplacians on a Directed Graph

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’s assumption. Moreover we establish isoperi-
metric inequalities in terms of the numerical range to show the lack of es-
sential spectrum of Laplacian on heavy ends directed graphs. We introduce
a special self-adjoint operator and compare its essential spectrum with that
of the non self-adjoint Laplacian considered.