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 $k$?

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# Lovasz theta function and independence number of product of simple odd-cycles

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 so, is there a bound for $\theta(C_{2k+1}) - \Theta(C_{2k+1})$ in general?