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Let $G$ be a graph and $\bar{G}$ its complement. Let $\boxtimes$ denote strong product. Let $\alpha(G)$ denote independence number of $G$ and $N_{G}$ be the number of vertices of $G$.

Is $$\left|\alpha\left({G \boxtimes \bar{G}}\right) - \alpha\left({\overline{G \boxtimes \bar{G}}}\right)\right| = O(\log(N_{G}))?$$

Do self-complementary graphs satisfy this property?

Atleast is there a family of graphs that satisfy this property?

Do there exist a family of graphs with the property: $$\left|\alpha\left({G \boxtimes \bar{G}}\right) - \alpha\left({\overline{G \boxtimes \bar{G}}}\right)\right| = O(\log(N_{G}))$$ where $G$ is the graph, $\bar{G}$ is its complement, $\boxtimes$ denotes the strong product, $\alpha(G)$ denotes the independence number of $G$ and $N_{G}$ is the number of vertices of $G$?