The essential spectrum of the Laplacian on rapidly branching tessellations

TL;DR

This paper characterizes when the essential spectrum of the Laplacian is empty for certain graphs, using hyperbolicity and curvature conditions, advancing understanding of spectral properties in geometric graph theory.

Contribution

It provides new criteria for the emptiness of the essential spectrum based on hyperbolicity and curvature for general graphs and planar tessellations.

Findings

Essential spectrum emptiness characterized by hyperbolicity.

Curvature-based criteria for planar tessellations.

Advances spectral analysis in geometric graph theory.

Abstract

In this paper we characterize emptiness of the essential spectrum of the Laplacian under a hyperbolicity assumption for general graphs. Moreover we present a characterization for emptiness of the essential spectrum for planar tessellations in terms of curvature.