The talk will consider ways of bounding the complexity of a graph as measured by the number of partitions satisfying certain properties. The approach adopted uses Vapnik Chervonenkis dimension techniques. An example of such a bound was given by Kleinberg et al (2004) with an application to network failure detection. We describe a new bound in the same vein that depends on the eigenvalues of the graph Laplacian. We show an application of the result to transductive learning of a graph labelling from examples.