# On Shannon Capacity

Let $G$ be a graph whose Shannon Capacity is $\Theta(G)$. Is there any graph product for which the Shannon Capacity is $\Theta(G)^k$ where $k$ is the number of times the product is taken?

The Shannon capacity is defined as $$\Theta(G) = \sup_k \alpha(G^k)^{1/k},$$ where $G^k$ is the strong graph product. It is not hard to show that $\alpha(G^{k+l}) \geq \alpha(G^k) \alpha(G^l)$, and this implies that $\alpha(G^k)^{1/k}$ converges. Therefore $$\Theta(G) = \lim_{k\to\infty} \alpha(G^k)^{1/k} = \lim_{k\to\infty} \alpha(G^{tk})^{1/{tk}} = \Theta(G^t)^{1/t}.$$