Weak convergence of finite graphs, integrated density of states and a Cheeger type inequality

In [G. Elek, On limits of finite graphs, Combinatorica, in press, URL: http://www.arxiv.org/pdf/math.CO/0505335] we proved that the limit of a weakly convergent sequence of finite graphs can be viewed as a graphing or a continuous field of infinite graphs. Thus one can associate a type II1-von Neumann algebra to such graph sequences. We show that in this case the integrated density of states exists, that is, the weak limit of the spectra of the graph Laplacians of the finite graphs is the KNS-spectral measure of the graph Laplacian of the limit graphing. Using this limit technique we prove a Cheeger type inequality for finite graphs.