I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM to partition them.For real data, I don't know the structure or pattern of the graph (the number of clusters, the diameters of the clusters, etc.).When I partition the graph into 2 clusters(by the sign of the eigenvector corresponding to second largest eigenvalue and without using kmeans or FCM) the result seems to be ok (the connections inside the cluster are strong) and it is natural because it is the global optimum, but for more than 2 I don't know how to determine if the result is good or not. Is there any optimum global solution for more than 2 clusters? How can I obtain the best result(with some mathematical reasoning)? any suggestion? I also ask similar question in math.st

Thanks.