Lovasz theta function and regular graphs (odd cycles in particular) - connections to spectral theory The post is related to:
https://mathoverflow.net/questions/59631/lovasz-theta-function-and-independence-number-of-product-of-simple-odd-cycles

How far away is the Lovasz bound from the zero-error capacity of regular graphs? Are there any examples where the Lovasz bound is known to be not equal to the zero-error capacity of a regular graph? (This was answered below by Oleksandr Bondarenko.)

In particular is any strict inequality known for odd cycles of sides greater than or equal to $7$?

Update
What improvement is needed in spectral theory to improve the Lovasz theta function so that the gap between Shannon capacity and Lovasz Theta for the cases for which a gap exists could be lowered? (Note I am concerned only from spectral perspective)
In fact there is known a regular graph $G$ for which zero-error capacity  $\Theta(G)$ is less than the Lov$\acute{a}$sz bound $\vartheta(G)$. W. Haemers in $[1]$ has proved that for the complement of Schl$\ddot{a}$fli graph $G$ the following holds: $\Theta(G)\leqslant7&lt;\vartheta(G)=9$.

In $[2]$ it is noted that "The best known upper bounds on $\Theta(C_m)$ and $\Theta(\overline{C}_m)$ for $m$ odd and greater than $5$ are given by the Lovasz theta function ...". From this I conclude that that the answer to your last question is no (since then I don't know any results improving on this.).

Finding Shannon capacity even for $C_7$ would be a major breakthrough for this hard problem. Additionally, it can be noticed that


  "it is not known if the problem of deciding whether the Shannon capacity of a given input graph exceeds a given value is decidable".



W. Haemers, “On some problems of Lov$\acute{a}$sz concerning the Shannon capacity of a graph,” IEEE Trans. on Information Theory, vol. 25, no. 2, pp. 231–232, Mar. 1979.
T. Bohman, "A limit theorem for the Shannon capacities of odd cycles. II," Proceedings of the American Mathematical Society, 2005.
N. Alon, "Combinatorial Reasoning in Information Theory".