  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.
