The dual Cheeger constant and spectra of infinite graphs

TL;DR

This paper introduces the dual Cheeger constant for infinite graphs, demonstrating its role in bounding the spectrum's top and characterizing the essential spectrum's behavior.

Contribution

It presents the dual Cheeger constant and establishes its significance in spectral analysis of infinite graphs, extending Cheeger theory to the spectrum's top.

Findings

Dual Cheeger constant bounds the top of the spectrum.

Dual Cheeger constant characterizes the shrinking of the essential spectrum.

The dual Cheeger constant at infinity indicates spectrum concentration.

Abstract

In this article we study the top of the spectrum of the normalized Laplace operator on infinite graphs. We introduce the dual Cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the Cheeger constant controls the bottom of the spectrum. Moreover, we show that the dual Cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized Laplace operator shrinks to one point.