Lovász Theta for self-complementary graphs

Question

Lovász (1979) in On the Shannon Capacity of Graphs write: If G is self-complementary, then $\Theta(G^2)=\Theta(G)^2$. Why?

Answer 1

The identity $\Theta(G^2) = \Theta(G)^2$ is true for any graph, not just a self-complementary one. It's only for the first half of the equation you're reading (which says $\Theta(G \cdot \overline{G}) = \Theta(G^2)$) that we need $G$ to be self-complementary.

(I think it's more modern to use $c(G)$ for Shannon capacity, but in this answer I'll stick to the notation used by Lovász.)

The proof is just some manipulation of limits: \begin{align} \Theta(G^2) &= \lim_{k\to\infty} \sqrt[k]{\alpha((G^2)^k)} \\ &= \lim_{k \to \infty} \sqrt[k]{\alpha(G^{2k})} \\ &= \left(\lim_{k \to \infty} \sqrt[2k]{\alpha(G^{2k})}\right)^2 \\ &= \left(\lim_{2k \to \infty}\sqrt[2k]{\alpha(G^{2k})}\right)^2 \\ &= \Theta(G)^2. \end{align} The only nontrivial step is that $(G^2)^k = G^{2k}$, which holds because graph products are associative.