Is anything known about Lovasz Theta Function taking integral value in non-perfect graphs? In particular, does integral value of Lovasz theta always coincide with the size of largest independent set?

For instance, graph below is non-perfect, and its Lovasz theta function gives the independence number.

Its true for a few other non-perfect graphs I tried, here's a Mathematica package I used to compute it

Is anything known about Lovasz Theta Function taking integral value in non-perfect graphs? In particular, does integral value of Lovasz theta always coincide with the size of largest independent set?

For instance, graph below is non-perfect, and its Lovasz theta function gives the independence number.

     (source)

Its true for a few other non-perfect graphs I tried, here's a Mathematica package I used to compute it

Is anything known about Lovasz Theta Function taking integral value in non-perfect graphs? In particular, does integral value of Lovasz theta always coincide with the size of largest independent set?

For instance, graph below is non-perfect, and its Lovasz theta function gives the independence number.

Is anything known about Lovasz Theta Function taking integral value in non-perfect graphs? In particular, does integral value of Lovasz theta always coincide with the size of largest independent set?

For instance, graph below is non-perfect, and its Lovasz theta function gives the independence number.

Its true for a few other non-perfect graphs I tried, here's a Mathematica package I used to compute it