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 $k$?



Lovasz theta function and independence number of product of simple odd-cycles
The theta function of odd cycles can be calculated explicitly:
$$\vartheta(C\_{2n+1})=\frac{(2n+1)\cos(\frac{\pi}{2n+1})}{1+\cos(\frac{\pi}{2n+1})}=n+\frac{1}{2}-O(1/n)$$
while computing $\Theta(C\_{2n+1})$ for any odd $2n+1\geq 7$ is an open problem. So your question is about bounds on $\Theta(C\_{2n+1})$. The best upper bound is the one we know for all graphs, $\Theta(G)\le \vartheta(G)$, there haven't been any improvements when $G$ is an odd cycle.


As for lower bounds, a first improvement on $\Theta(C\_{2n+1})\geq \alpha(C\_{2n+1})=n$ is given by
$$\Theta(C\_{2n+1})\geq \sqrt{\alpha(C\_{2n+1}^2)}=\sqrt{n^2+\lfloor\frac{n}{2}\rfloor}=n+\frac{1}{4}-O(1/n)$$
and further by considering a lower bound on $\alpha(C\_{2n+1}^3)$, Bohman, Ruszink and Thoma proved in "Shannon capacity of large odd cycles" that $$\Theta(C\_{2n+1})\geq n+\frac{1}{3}-O(1/n)$$
I believe the best known lower bounds are the ones one gets from extracting the exact counting given in ["A limit theorem for the Shannon capacities of odd cycles, I"](http://www.jstor.org/pss/1194666) and II, by T. Bohman. Bohman didn't write the explicit lower bound but instead just gave an estimate which is enough to prove
$$\lim\_{n\to \infty} \left(n+\frac{1}{2}-\Theta(C\_{2n+1})\right)=0$$


