G-Graphs and Algebraic Hypergraphs

G-graphs have been introduced in [A. Bretto, A. Faisant. A new way for associating a graph to a group, Mathematica Slovaca 55, No.1, (2005), 1-8], and have some applications in both symmetric and semi-symmetric graphs, cage graphs, and expander graphs [A. Bretto, A. Faisant and L. Gillibert, G-graphs: A new representation of groups. Journal of Symbolic Computation, Volume 42, Issue 5, May 2007, Pages 549-560], [A. Bretto, L. Gillibert and B. Laget, Symmetric and Semisymmetric Graphs Construction Using G-graphs, Proceedings of ISSAC'05, July 24-27 2005, Beijing, China. ACM Proceedings (2005), 61-67], [A. Bretto, L. Gillibert, A New Upper Bound for the (p,6) cages, Proceedings of ISSAC'07, July 29-August 1st 2007, Waterloo Canada. ACM Proceedings (2007)]. In this paper we present a characterization of G-graphs, introduce the notion of principal clique hypergraphs, and present some of their basic properties.