# NP-hardness on Cayley graphs

What is known about complexity of NP-hard problems on Cayley graphs?

Suppose that the graph is given explicitly as the multiplication table of the group and the list of generators. So the input length is the size of the graph. Can we solve NP-complete problems on such graphs (maximum clique/max-cut) in polynomial time?

What about some special cases of groups? For example, $\mathbb{Z}_n$ (a.k.a. circulant graphs) or $\mathbb{Z}_2^{\log(n)}$. That is, the input to the problem is the set of generators (and $1^n$ to represent the size of the graph).