In this paper we propose a novel graph-based genetic algorithm for the evolution of novel molecular graphs from a predefined set of elements or molecular fragments with an external objective function. A brief overview of existing genetic algorithm approaches in molecular design is provided followed by a description of our approach. The paper continues to suggest a novel application of this program to the multiobjective evolution of median molecules that are structurally representative of a set of objective molecules. We conclude with a summary of our initial results along with a discussion of a variety of improvements and applications of our approach.