In this talk, we will discuss graph cores, graph clustering and their application to a real problem. A core in a graph is usually taken as a set of highly connected vertices. Although general, this definition is intuitive and useful for studying the structure of many real networks. Nevertheless, depending on the problem, different formulations of graph core may be required, leading us to the known concept of generalized core. Thus, we study and further extend the notion of generalized core. Given a graph, we propose a definition of graph core based on a subset of its subgraphs and on a subgraph property function. Our approach generalizes several notions of graph core proposed independently in the literature, introducing a general and theoretical sound framework for the study of fully generalized graph cores. Moreover, we discuss emerging applications of graph cores, such as improved graph clustering methods and complex network motif detection. In particular, we discuss an application to query log mining.