Graphical and computational representation of groups Alain Bretto 1 and Luc Gillibert 1 Universit´ e de Caen, GREYC CNRS UMR-6072,Campus II, Bd Marechal Juin BP 5186, 14032 Caen cedex, France. alain.bretto@info.unicaen.fr, lgillibe@info.unicaen.fr Abstract. An important part of the computer science is focused on the links that can be established between group theory and graph theory. Cayley graphs can establish such a link but meet some limitations. This paper introduces a new type of graph associated to a group: the G-graphs. We present an implementation of the algorithm constructing these new graphs. We establish a library of the most common G-graphs, using GAP and the SmallGroups library. We give some experimental results with GAP and we show that many classical graphs are G-graphs. 1 Introduction The group theory, especially the finite group theory, is one of the main parts of modern mathematics. Groups are objects designed for the study of symmetries and symmetric structures, and therefore many sciences have to deal with them. Graphs can be interesting tools for the study of groups, a popular representa- tion of groups by graphs being the Cayley graphs, an extended research has been achieved in this direction [1]. The regularity and the underlying algebraic structure of Cayley graphs make them good candidates for applications such as optimizations on parallel architectures, or for the study of interconnection networks [4]. But these properties are also a limitation: many interesting graphs are not Cayley graphs. The purpose of this paper is to introduce a new type of graph – called G-graphs – constructed from a group and to present an algorithm to construct them. This al- gorithm is used for establishing some experimental results and for finding which graphs are G-graphs and which graphs are not. In fact, G-graphs, like Cay- ley graphs, have both nice and highly-regular properties. Consequently, these graphs can be used in any areas of science where Cayley graphs occur. Moreover many usual graphs, as the cube, the hypercube, the cuboctahedral graph, the Heawood’s graph and lots of others, are G-graphs. We prove that some generic and infinite families of graphs, such as the complete bipartite graphs, are G- graphs. We establish a catalogue of the most common G-graphs, and for each of these graphs we exhibit the corresponding group, using the GAP’s SmallGroups library. We also show that some non-vertex-transitive graphs, such as the Gray graph and the Ljubljana graph, are also G-graphs. In contrast, notice that Cay- ley graphs are always vertex-transitive.