Essential spectrum of the discrete Laplacian on a perturbed periodic graph

TL;DR

This paper investigates the stability of the essential spectrum of the Laplacian on perturbed periodic graphs, including non-periodic cases, and provides methods to determine spectra of various complex graph structures.

Contribution

It introduces a class of perturbed graphs with stable essential spectra despite infinite modifications, extending spectral analysis to non-periodic and randomly altered graphs.

Findings

Essential spectra remain stable under certain infinite perturbations.

Methods to compute spectra of cone-like and random vertex addition graphs.

Extension of spectral stability results to non-periodic graph perturbations.

Abstract

We address the Laplacian on a perturbed periodic graph which might not be a periodic graph. We present a class of perturbed graphs for which the essential spectra of the Laplacians are stable even when the graphs are perturbed by adding and removing infinitely many vertices and edges. Using this result, we demonstrate how to determine the spectra of cone-like graphs, the upper-half plane, and graphs obtained from $\mathbb{Z}^2$ by randomly adding vertices.