Symmetry and connectivity in G-graphs

This article presents some interesting properties about a new type of graph associated to a group, the G-graphs [A. Bretto and L. Gillibert. Graphical and computational representation of groups, LNCS 3039, Springer-Verlag pp 343-350. Proceedings of ICCSʹ2004]. We show that many properties of a group can be seen on its associated G-graph and that many common graphs are G-graphs. We explain how to build efficiently some symmetric and semisymmetric graphs using the G-graphs. We establish a link beetwen Cayley graphs [A. Cayley. The theory of groups: graphical representations. Amer. J. of Math., 1878, 1:174–176; A. Cayley. On the theory of groups. Amer. J. of Math, 1889, 11:139–157] and G-graphs.