TITLE: Lovasz theta function integrality
QUESTION [8 upvotes]: 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

REPLY [4 votes]: No. For some examples see When the Lovász theta-function saturates its upper bound