  Graphs derived from groups are a widely studied class of graphs, motivated by
their highly symmetric structure. In particular, G-graphs offer an easy and
interesting alternative construction of semi-symmetric graphs. After recalling
the main properties of these graphs, this papers gives an extended
characterisation of G-graphs and develops the link between bipartite G-graphs
and incidence graphs. It appears that these two classes of graphs have a wide
overlapping despite having completely different constructions. We give partial
answers to the problem of finding which complete simple graphs have a G-graph
as their incidence graph.
