This Thesis Is Submitted to the University of Ghana, Legon in Partial Fulfilment of the Requirements for the Award of Master of Philosophy Mathematics Degree

The G-graph Γ(G,S) is a graph from the group G generated by S ⊆ G, where the vertices are the right cosets of the cyclic subgroups 〈s〉, s ∈ S with k-edges between two distinct cosets if there is an intersection of k elements. In this thesis, after presenting some important properties of G-graphs, we show how the G-graph depends on the generating set of the group. We give the G-graphs of the symmetric group, alternating group and the semi-dihedral group with respect to various generating sets. We give a characterisation of nite G-graphs; in the general case and a bipartite case. Using these characterisations, we give several classes of graphs that are G-graphs. For instance, we consider the Turán graphs, the platonic graphs and biregular graphs such as the Levi graphs of geometric con gurations. We emphasis the structural properties of G-graphs and their relations to the group G and the generating set S. As preliminary results for further studies, we give the adjacency matrix and spectrum of various nite G-graphs. As an application, we compute the energy of these graphs. We also present some preliminary results on in nite G-graphs where we consider the G-graphs of the in nite group SL2(Z) and an in nite non-Abelian matrix group.