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We show that for any graph $G$, by considering "activation" through thestrong product with another graph $H$, the relation $\alpha(G) \leq\vartheta(G)$ between the independence number and the Lov\'{a}sz number of $G$can be made arbitrarily tight: Precisely, the inequality \[  \alpha(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H) \]becomes asymptotically an equality for a suitable sequence of ancillary graphs$H$.  This motivates us to look for other products of graph parameters of $G$ and$H$ on the right hand side of the above relation. For instance, a result ofRosenfeld and Hales states that \[  \alpha(G \times H) \leq \alpha^*(G)\,\alpha(H), \] with the fractionalpacking number $\alpha^*(G)$, and for every $G$ there exists $H$ that makes theabove an equality; conversely, for every graph $H$ there is a $G$ that attainsequality.  These findings constitute some sort of duality of graph parameters, mediatedthrough the independence number, under which $\alpha$ and $\alpha^*$ are dualto each other, and the Lov\'{a}sz number $\vartheta$ is self-dual. We also showduality of Schrijver's and Szegedy's variants $\vartheta^-$ and $\vartheta^+$of the Lov\'{a}sz number, and explore analogous notions for the chromaticnumber under strong and disjunctive graph products.

A new property of the Lov\'asz number and duality relations between graph parameters