In this paper, a new graph based framework for clustering characterization is proposed. In this context, Self Organizing Map (SOM) is one popular method for clustering and visualizing high dimensional data, which is generally succeeded by another clustering methods (partitional or hierarchical) for optimizing the final partition. Recently, we have developed a new SOM clustering method based on graph coloring called McSOM. In the current study, we propose to automatically characterize the classes obtained by this method. To this end, we propose a new approach combining a statistical test with a maximum spanning tree for local features selection in each class. Experiments will be given over several databases for validating our approach.