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)