Is there an evidence or a proof that the independence of strong products of graphs can increase after stabilizing?I am interested in odd cycles only.Let $C_n$ be an odd cycle and $\alpha(G)$ represent independence number of a graph $G$. Let $C_n^m$ be $m$-times strong product of $C_n$.Now say $$\alpha(C_n^{m_1m_2})=\alpha(C_n^{m_1})^{m_2}.$$ That is $m_1m_2$-times strong product of $C_n$ has same independence number as $m_1$-times strong product of $C_n$ raised to $m_2$. For example,  $\alpha(C_5^{4})=\alpha(C_5^{2})^{2}=25$. Then is it possible $$\alpha(C_n^{m_1m_2m_3})>\alpha(C_n^{m_1})^{m_2m_3}\mbox{ or }\alpha(C_n^{m_1m_2m_3})>\alpha(C_n^{m_1m_2})^{m_3}?$$In the example above, we have violation of strict inequality since we have:$$\alpha(C_5^{8})=\alpha(C_5^{2})^{4}=625\mbox{ or }\alpha(C_5^{8})=\alpha(C_5^{4})^{2}=625.$$
Can independence numbers of box products of cycles increase after stabilizing?
graph theory;information theory