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
the MathML source">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.