A convex quadratic characterization of the Lovász theta number

In previous works an upper bound on the stability number $\alpha(G)$ of a graph G based on convex quadratic programming was introduced and several of its properties were established. The aim for this investigation is to relate theoretically this bound (usually represented by $\upsilon(G)$) with the well-known Lovász $\vartheta(G)$ number. First, a new set of convex quadratic bounds on $\alpha(G)$ that generalize and improve the bound $\upsilon(G)$ is proposed. Then it is proved that $\vartheta(G)$ is never worse than any bound belonging to this set of new bounds. The main result of this note states that one of these new bounds equals $\vartheta(G)$, a fact that leads to a new characterization of the Lovász theta number.