This paper introduces a general framework for defining the entropy of a graph. Our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph. More precisely, an information functional quantifies structural information of a graph based on a derived probability distribution. Such a probability distribution leads directly to an entropy of a graph. Then, the structural information content of a graph will be is interpreted and defined as the derived graph entropy. Another major contribution of this paper is the investigation of relationships between graph entropies. In addition to this, we provide numerical results demonstrating not only the feasibility of our method, which has polynomial time complexity, but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks.