In this work, we present a graph-based method for the detection and classification of malicious software samples utilizing the Max-Flows exhibited through their corresponding behavioral graphs. In the proposed approach, we utilize the Max-Flows exhibited in the behavioral graphs that represent the interaction of software samples with their host environment, in order to depict the flow of information between System-call Groups. Obtaining the System-call Dependency Graphs of the samples under consideration, we construct the corresponding Group Relation Graphs, and proceed with the construction of the so-called, Flow Maps, another representation of Group Relation Graphs, that depict the Max-Flows among its vertices. Additionally, we provide a detailed representation over the architecture and the core components of our proposed approach for malware detection and classification discussing also several technical aspects regarding its implementation and deployment. Finally, we conduct a series of five-fold cross validation experiments in order to evaluate the potentials of our proposed approach in detecting and classifying malicious samples discussing also the exhibited experimental results.