Spectrum of the Laplacian on a covering graph with pendant edges I: The one-dimensional lattice and beyond

TL;DR

This paper investigates the spectral properties of Laplacians on covering graphs derived from integer lattices with pendant edges, revealing conditions for spectral gaps and eigenvalues in one and two dimensions.

Contribution

It provides a detailed analysis of the spectral behavior of Laplacians on such graphs, including conditions for the existence of spectral gaps and eigenvalues, extending understanding beyond standard lattices.

Findings

Laplacian on 1D lattice with pendant edges has a spectral gap.

Conditions are established for when the Laplacian has no eigenvalues.

In 2D, certain arrangements of pendant edges eliminate the spectral gap.

Abstract

In this paper, we examine covering graphs that are obtained from the $d$-dimensional integer lattice by adding pendant edges. In the case of $d=1$, we show that the Laplacian on the graph has a spectral gap and establish a necessary and sufficient condition under which the Laplacian has no eigenvalues. In the case of $d=2$, we show that there exists an arrangement of the pendant edges such that the Laplacian has no spectral gap.