An important part of computer science is focused on the links that can be established between group theory and graph theory and graphs. CAYLEY graphs, that establish such a link, are useful in a lot of areas of sciences. This paper introduces a new type of graph associated with a group, the G-graphs, and presents many of their properties. We show that various characteristics of a group can be seen on its associated G-graph. We also present an implementation of the algorithm constructing these new graphs, an implementation that will lead to some experimental results. Finally we show th