We propose a new method for detecting communities based on the concept of communicability between nodes in a complex network. This method, designated as N-ComBa K-means, uses a normalized version of the adjacency matrix to build the communicability matrix and then applies K-means clustering to find the communities in a graph. We analyze how this method performs for some pathological cases found in the analysis of the detection limit of communities and propose some possible solutions on the basis of the analysis of the ratio of local to global densities in graphs. We use four different quality criteria for detecting the best clustering and compare the new approach with the Girvan-Newman algorithm for the analysis of two "classical" networks: karate club and bottlenose dolphins. Finally, we analyze the more challenging case of homogeneous networks with community structure, for which the Girvan-Newman completely fails in detecting any clustering. The N-ComBa K-means approach performs very well in these situations and we applied it to detect the community structure in an international trade network of miscellaneous manufactures of metal having these characteristics. Some final remarks about the general philosophy of community detection are also discussed.