Lovasz theta function $\theta(G)$ of a graph $G$ provides an upper bound for the independence number of a graph, $\alpha(G)$ and $\Theta(G) = \lim_{k\rightarrow \infty}\sqrt[k]{\alpha(G^{k})}$. That is, $\Theta(G) \le \theta(G)$.
If the graph is a pentagon ($G=C_{5}$), then $\Theta(C_{5}) = \theta(C_{5})$.
Is
$\Theta(C_{2k+1}) \ne \theta(C_{2k+1})$ if $k > 2$ ? If sosince $\theta(C_{2k+1})^{r}$ fails to be integral for any $r \in \mathbb{Z}^{+}$. However, is are there a bound known lower and upper bounds for for $\theta(C_{2k+1}) - \Theta(C_{2k+1})$ in general?that depends on $k$?
