Let $G$ be a graph of maximum degree $d$. Are there any known/easy bounds on the clique number of $G^k$, in terms of $d$ and $k$? I would like something at least epsilon better than the bound on the degree of $G^k$ (something like half of this bound would be very nice, and not surprising), and am particularly interested in the case $d=4$ and $k=4$.

EDIT: Apparently this is well-studied, and called the degree-diameter problem. So here is a twist: what if some vertex is in at least two triangles?