Shannon capacity $\Theta(G)$ of pentagon is achieved at $2$-fold strong product of the pentagon.


It is also known that the Lov\'asz theta $\vartheta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite positive integer $m$ if $G$ is an odd cycle of length $>5$.


**Is it known that $\Theta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite positive integer $m$ if $G$ is an odd cycle of length $>5$? That is $\limsup\_{m\rightarrow\infty}\alpha(G^{\boxtimes m})^{\frac{1}{m}}$ is not attained at a finite positive integer $m$.**


Note that the the above statement is true would still not be strong enough to decide $\vartheta(G)=\Theta(G)$.



Capacity of Cycle Graphs
Currently, there are no known examples of graphs where the Shannon Capacity is realized in a dimension other than $1$ or $2$.


However, there are cases (Such as $C\_5$ union a point) where the Shannon Capacity is not achieved in any dimension, and it is believed by many that this is the case for $C\_7$.


