Graph cut is a common way of clustering nodes on similarity graphs. As a clustering method, it does not give a unique solution under usually used loss functions. We specifically show the problem in similarity graph-based clustering setting that the resulting clusters might be even disconnected. This is counter-intuitive as one wish to have good clustering solutions in the sense that each cluster is well connected and the clusters are balanced. The key property of good clustering solutions is that the resulting graphs (after clustering) share large components with the original ones. We wish to detect this case by deriving a graph similarity measure that shows high similarity values to the original graph for good clustering solutions. The similarity measure considers global connectivities of graphs by treating graphs as distributions in (potentially different) Euclidean spaces. The global graph comparison is then turned into distribution comparison. Simulation shows that the similarity measure could consistently distinguish different qualities of clustering solution beyond what could be done with the usually used loss functions of clustering algorithms.