Lovasz theta function $\vartheta(G)$ 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, $\Theta(G) \le \vartheta(G)$.

If the graph is a pentagon ($G=C_{5}$), then $\Theta(C_{5}) = \vartheta(C_{5})$.

$\Theta(C_{2k+1}) \ne \vartheta(C_{2k+1})$ if $k > 2$ since $\vartheta(C_{2k+1})^{r}$ fails to be integral for any $r \in \mathbb{Z}^{+}$. However, are there known lower and upper bounds for for $\vartheta(C_{2k+1}) - \Theta(C_{2k+1})$ that depends on $k$?