Lord Kavi, University of Ottawa
Structural Properties and Characterisation of finite G-graphs

The G-graph Gamma(G,S) of a group G with generating set S is the graph whose vertices are the right cosets of the cyclic subgroups generated by the individual elements of S, with two vertices joined by k parallel edges if the corresponding cosets intersect in exactly k elements. In this talk, after presenting some 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 finite G-graphs, both in the bipartite case and the general case. Using these characterisations, we give several classes of graphs that are G-graphs. For instance, we consider the Turan graphs, the platonic graphs, and biregular graphs, such as the Levi graphs of geometric configurations. We emphasise structural properties of G-graphs and their relationship to the group G and the generating set S. We also present some preliminary results on infinite G-graphs, where we consider the G-graphs of the infinite group SL_2(Z) and an infinite non-Abelian matrix group.