For a finite group $$G$$ and a non empty subset of generators $$S \subseteq G$$, $$\phi(G,S)=(V,E)$$ is a $$G$$-graph where $$V$$ is the union of the sets of $$(x, sx, s^2 x, ... , s^{o(s)-1}x)$$ for all $$s \in S$$. Two distinct vertices $$(s)x$$ and $$(t)y$$ are adjacent iff $$|<s>x \cap <t>y | \geq 1$$.
Bretto et. al defined the $$G$$-graph in 2005 in order to answer some conjectures in graph theory like graph isomorphism and cage problem. Here I will present my results on Lovasz' conjecture (1970) and find the Hamiltonian Cayley graphs.